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Metamath Proof Explorer

Theorem dfcleq

Description: The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality ( ax-ext ) and the definition of class equality ( df-cleq ). Its forward implication is called "class extensionality". Remark: the proof uses axextb to prove also the hypothesis of df-cleq that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen , equid }. (Contributed by NM, 15-Sep-1993) (Revised by BJ, 24-Jun-2019)

Ref Expression
Assertion dfcleq ⊢ A = B ↔ ∀ x x ∈ A ↔ x ∈ B

Proof

Step Hyp Ref Expression
1 axextb ⊢ y = z ↔ ∀ u u ∈ y ↔ u ∈ z
2 axextb ⊢ t = t ↔ ∀ v v ∈ t ↔ v ∈ t
3 1 2 df-cleq ⊢ A = B ↔ ∀ x x ∈ A ↔ x ∈ B