Theorem eldif

Description: Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	eldif	⊢ ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) → 𝐴 ∈ V )
2		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
3	2	adantr	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) → 𝐴 ∈ V )
4		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
5		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶 ) )
6	5	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝑥 ∈ 𝐶 ↔ ¬ 𝑦 ∈ 𝐶 ) )
7	4 6	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) ) )
8		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
9		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
10	9	notbid	⊢ ( 𝑦 = 𝐴 → ( ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶 ) )
11	8 10	anbi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) ) )
12		df-dif	⊢ ( 𝐵 ∖ 𝐶 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) }
13	7 11 12	elab2gw	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) ) )
14	1 3 13	pm5.21nii	⊢ ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) )