Theorem elin

Description: Expansion of membership in an intersection of two classes. Theorem 12 of Suppes p. 25. (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ V )
2		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
3	2	adantl	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → 𝐴 ∈ V )
4		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
5		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶 ) )
6	4 5	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
7		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
8		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
9	7 8	anbi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) ) )
10		df-in	⊢ ( 𝐵 ∩ 𝐶 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) }
11	6 9 10	elab2gw	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) ) )
12	1 3 11	pm5.21nii	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )