Theorem dfss4 2238

Description: Subclass defined in terms of class difference. See comments under dfun2 2239.

Assertion

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

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 2225	. 2 ⊢ (A ⊆ B ↔ (B ∩ A) = A)
2		abai 479	. . . . . 6 ⊢ ((x ∈ B ⋀ x ∈ A) ↔ (x ∈ B ⋀ (x ∈ B → x ∈ A)))
3		iman 237	. . . . . . 7 ⊢ ((x ∈ B → x ∈ A) ↔ ¬ (x ∈ B ⋀ ¬ x ∈ A))
4	3	anbi2i 480	. . . . . 6 ⊢ ((x ∈ B ⋀ (x ∈ B → x ∈ A)) ↔ (x ∈ B ⋀ ¬ (x ∈ B ⋀ ¬ x ∈ A)))
5	2, 4	bitr 173	. . . . 5 ⊢ ((x ∈ B ⋀ x ∈ A) ↔ (x ∈ B ⋀ ¬ (x ∈ B ⋀ ¬ x ∈ A)))
6		elin 2203	. . . . 5 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ⋀ x ∈ A))
7		eldif 2053	. . . . . 6 ⊢ (x ∈ (B ∖ (B ∖ A)) ↔ (x ∈ B ⋀ ¬ x ∈ (B ∖ A)))
8		eldif 2053	. . . . . . . 8 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ⋀ ¬ x ∈ A))
9	8	negbii 187	. . . . . . 7 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ⋀ ¬ x ∈ A))
10	9	anbi2i 480	. . . . . 6 ⊢ ((x ∈ B ⋀ ¬ x ∈ (B ∖ A)) ↔ (x ∈ B ⋀ ¬ (x ∈ B ⋀ ¬ x ∈ A)))
11	7, 10	bitr 173	. . . . 5 ⊢ (x ∈ (B ∖ (B ∖ A)) ↔ (x ∈ B ⋀ ¬ (x ∈ B ⋀ ¬ x ∈ A)))
12	5, 6, 11	3bitr4 183	. . . 4 ⊢ (x ∈ (B ∩ A) ↔ x ∈ (B ∖ (B ∖ A)))
13	12	eqriv 1472	. . 3 ⊢ (B ∩ A) = (B ∖ (B ∖ A))
14	13	eqeq1i 1479	. 2 ⊢ ((B ∩ A) = A ↔ (B ∖ (B ∖ A)) = A)
15	1, 14	bitr 173	1 ⊢ (A ⊆ B ↔ (B ∖ (B ∖ A)) = A)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 954   ∈ wcel 956   ∖ cdif 2040   ∩ cin 2042   ⊆ wss 2043

This theorem is referenced by:  dfin4 2244  sbthlem3 4435  isopn2 7623  iincld 7629  ntrval2 7636  cmclsopn 7643  cmntrcld 7644  islp2 7697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049

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