x ∈ F  ⇔ {x} ⊂ F ⇔ (∃y∈E, x=y ∧ y∈F) ⇔ (∀y∈E, x=y ⇒ y∈F) 
x ∈ F  ⇒ ((∀y∈F, A(y)) ⇒ A(x) ⇒ ∃y∈F, A(y)) 
F ⊂ {x}  ⇔ (∀y∈F, x=y) ⇒ ((∃y∈F, A(y)) ⇒ A(x) ⇒ (∀y∈F, A(y))) 
F={x}  ⇔ (x∈F ∧ ∀y∈F, x=y) ⇔ (∀y∈E, y∈F ⇔ x=y) 
Here are 3 new quantifiers: ∃^{≥2} (plurality), ! (uniqueness), and ∃! (existence and uniqueness), whose results when applied to A in E only depend on F={x∈E  A(x)} (like ∃ and unlike ∀) :
(∃x∈E, A(x)) ⇔ (F ≠ ∅)  ⇔ (∃x∈F, 1) ⇔ (∃x∈E, {x} ⊂ F) 
(∃^{≥2}x∈E, A(x)) ⇔ (∃^{≥2}: F)  ⇔ (∃x,y∈F, x ≠ y) ⇔ (∃x≠y∈E, A(x) ∧ A(y)) 
(!x∈E, A(x)) ⇔ (!:F)  ⇔ ¬(∃^{≥2}: F) ⇔ (∀x,y∈F, x=y) ⇔ ∀x∈F, F ⊂ {x} 
(∃!x∈E, A(x)) ⇔ (∃!:F)  ⇔ (∃x∈F, F⊂{x}) ⇔ (∃x∈E, F={x}) 
F ⊂ {x}  ⇒ (∀y∈F, F⊂{y}) ⇔ (!:F) 
(∃!:F)  ⇔ (F≠∅ ∧ !: F) 
F≠∅  ⇒ ((∀y∈F, B(y)) ⇒ (∃y∈F, B(y))) 
( !: F)  ⇒ ((∃y∈F, B(y)) ⇒ (∀y∈F, B(y))) 
F={x}  ⇒ ((∃y∈F, B(y)) ⇔ B(x) ⇔ ∀y∈F, B(y)) 
∀x, ℩{x} = x
∀_{Set}E,
∃!:E ⇒ ℩E ∈ E
(A ? B : C) ⇔
(¬C⇒A⇒B) ⇔ ((A⇒B)∧(¬A⇒C))
⇔ (¬A ? C : B)
⇔ ((A∧B)∨(¬A∧C)) ⇔((C⇒A)⇒(A∧B))
⇎ (A ? ¬B : ¬C)
¬A ⇔ (A ? 0 : 1)
(A⇒B) ⇔ (A
? B : 1)
(A∧B) ⇔ (A ? B : 0)
(A∨B) ⇔ (A ? 1: B)
(A ⇔ B) ⇔ (A ? B : ¬B).
(B ? x : y) = ℩{z∈{x,y} B ? z=x : z=y} = ℩{z∈{x,y} z=x ⇔ B}
so that for any unary predicate A,A(B ? x : y) ⇔ (B ? A(x) : A(y)).
All paraoperators other than connectives but with at least a Boolean argument, are naturally expressed as composed of the conditional operator with operators, or the conditional connective with predicates, which is why they did not need an explicit presence in the language of a theory.r = (E×F ∋ (x,y) ↦
(R(x,y) ? 1 : 0))
∀x∈E, ∀y∈F,
R(x,y) ⇔ r(x, y) = 1
(x,y) = (V_{2} ∋ a ↦ (a = 1 ? y : x))
Confusing Booleans with objects, (B ? x : y) = π_{B}(y,x).(x,y,z) = (V_{3} ∋ a ↦ (a = 2 ? z : (a = 1 ? y : x)))
Given the notion of ordered pairs, the roles of notions of tuples of any higher arity n > 2 can also be played by other classes, with other definitions of their definers and projections, satisfying the same axioms. For example, triples t = (x,y,z) can be defined as t = ((x,y),z)) and evaluated byx = π_{0}(π_{0}(t))
y = π_{1}(π_{0}(t))
z = π_{1}(t)
Set theory and Foundations of Mathematics  
1. First foundations of mathematics  
2. Set theory  
2.1.
First axioms of set theory 2.2. Set generation principle 2.3. Tuples ⇦ 2.4. Uniqueness quantifiers ⇨ 2.5. Families, Boolean operators on sets 2.6. Graphs 2.7. Products and powerset 
2.8. Injections, bijections 2.9. Properties of binary relations 2.10. Axiom of choice 
Time
in set theory Interpretation of classes Concepts of truth in mathematics 

3. Algebra  4. Arithmetic  5. Secondorder foundations 