Theorem elprg

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, generalized. (Contributed by NM, 13-Sep-1995)

	Ref	Expression
Assertion	elprg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐵 ↔ 𝑦 = 𝐵 ) )
2		eqeq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐶 ↔ 𝑦 = 𝐶 ) )
3	1 2	orbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ) ↔ ( 𝑦 = 𝐵 ∨ 𝑦 = 𝐶 ) ) )
4		eqeq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 = 𝐵 ↔ 𝐴 = 𝐵 ) )
5		eqeq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 = 𝐶 ↔ 𝐴 = 𝐶 ) )
6	4 5	orbi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 = 𝐵 ∨ 𝑦 = 𝐶 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) )
7		dfpr2	⊢ { 𝐵 , 𝐶 } = { 𝑥 ∣ ( 𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ) }
8	3 6 7	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) )