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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 ) )