Theorem dfss4 1667

Description: Subclass defined in terms of class difference. See comments under dfun2 1668.

Assertion

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

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 1656	. 2 ⊢ (A ⊆ B ↔ (B ∩ A) = A)
2		abai 366	. . . . . 6 ⊢ ((x ∈ B ∧ x ∈ A) ↔ (x ∈ B ∧ (x ∈ B → x ∈ A)))
3		iman 205	. . . . . . 7 ⊢ ((x ∈ B → x ∈ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
4	3	anbi2i 367	. . . . . 6 ⊢ ((x ∈ B ∧ (x ∈ B → x ∈ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
5	2, 4	bitr 151	. . . . 5 ⊢ ((x ∈ B ∧ x ∈ A) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
6		elin 1635	. . . . 5 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ x ∈ A))
7		eldif 1496	. . . . . 6 ⊢ (x ∈ (B ∖ (B ∖ A)) ↔ (x ∈ B ∧ ¬ x ∈ (B ∖ A)))
8		eldif 1496	. . . . . . . 8 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
9	8	negbii 162	. . . . . . 7 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
10	9	anbi2i 367	. . . . . 6 ⊢ ((x ∈ B ∧ ¬ x ∈ (B ∖ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
11	7, 10	bitr 151	. . . . 5 ⊢ (x ∈ (B ∖ (B ∖ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
12	5, 6, 11	3bitr4 158	. . . 4 ⊢ (x ∈ (B ∩ A) ↔ x ∈ (B ∖ (B ∖ A)))
13	12	cleqri 1101	. . 3 ⊢ (B ∩ A) = (B ∖ (B ∖ A))
14	13	cleq1i 1108	. 2 ⊢ ((B ∩ A) = A ↔ (B ∖ (B ∖ A)) = A)
15	1, 14	bitr 151	1 ⊢ (A ⊆ B ↔ (B ∖ (B ∖ A)) = A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  dfin4 1673  sbthlem3 3351

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492

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