Theorem dfss4 2232

Description: Subclass defined in terms of class difference. See comments under dfun2 2233.

Assertion

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

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 2219	. 2 ⊢ (A ⊆ B ↔ (B ∩ A) = A)
2		abai 478	. . . . . 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 479	. . . . . 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 2197	. . . . 5 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ⋀ x ∈ A))
7		eldif 2047	. . . . . 6 ⊢ (x ∈ (B ∖ (B ∖ A)) ↔ (x ∈ B ⋀ ¬ x ∈ (B ∖ A)))
8		eldif 2047	. . . . . . . 8 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ⋀ ¬ x ∈ A))
9	8	negbii 187	. . . . . . 7 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ⋀ ¬ x ∈ A))
10	9	anbi2i 479	. . . . . 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 1467	. . 3 ⊢ (B ∩ A) = (B ∖ (B ∖ A))
14	13	eqeq1i 1474	. 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 953   ∈ wcel 955   ∖ cdif 2034   ∩ cin 2036   ⊆ wss 2037

This theorem is referenced by:  dfin4 2238  sbthlem3 4429  isopn2 7615  iincld 7621  ntrval2 7628  cmclsopn 7635  cmntrcld 7636  islp2 7688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043

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