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 = 𝐵𝐴 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 elpr.1 𝐴 ∈ V
2 elprg ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐵𝐴 = 𝐶 ) ) )
3 1 2 ax-mp ( 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐵𝐴 = 𝐶 ) )