Theorem dfss4

Description: Subclass defined in terms of class difference. See comments under dfun2 . (Contributed by NM, 22-Mar-1998) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	dfss4	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sseqin2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 )
2		eldif	⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) )
3	2	notbii	⊢ ( ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ↔ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) )
4	3	anbi2i	⊢ ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) ) )
5		elin	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) )
6		abai	⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ) )
7		iman	⊢ ( ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ↔ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) )
8	7	anbi2i	⊢ ( ( 𝑥 ∈ 𝐵 ∧ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) ) )
9	5 6 8	3bitri	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) ) )
10	4 9	bitr4i	⊢ ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) ↔ 𝑥 ∈ ( 𝐵 ∩ 𝐴 ) )
11	10	difeqri	⊢ ( 𝐵 ∖ ( 𝐵 ∖ 𝐴 ) ) = ( 𝐵 ∩ 𝐴 )
12	11	eqeq1i	⊢ ( ( 𝐵 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 )
13	1 12	bitr4i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴 )