Metamath Proof Explorer

Theorem elpr

Description: A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of TakeutiZaring p. 15. (Contributed by NM, 13-Sep-1995)

Ref Expression
Hypothesis elpr.1 ⊢ A ∈ V
Assertion elpr ⊢ A ∈ B C ↔ A = B ∨ A = C

Proof

Step Hyp Ref Expression
1 elpr.1 ⊢ A ∈ V
2 elprg ⊢ A ∈ V → A ∈ B C ↔ A = B ∨ A = C
3 1 2 ax-mp ⊢ A ∈ B C ↔ A = B ∨ A = C