Theorem dfss4 3490

Description: Subclass defined in terms of class difference. See comments under dfun2 3491. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem dfss4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqin2 3475	. 2 ⊢ (A ⊆ B ↔ (B ∩ A) = A)
2		eldif 3222	. . . . . . 7 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
3	2	notbii 287	. . . . . 6 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
4	3	anbi2i 675	. . . . 5 ⊢ ((x ∈ B ∧ ¬ x ∈ (B ∖ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
5		elin 3220	. . . . . 6 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ x ∈ A))
6		abai 770	. . . . . 6 ⊢ ((x ∈ B ∧ x ∈ A) ↔ (x ∈ B ∧ (x ∈ B → x ∈ A)))
7		iman 413	. . . . . . 7 ⊢ ((x ∈ B → x ∈ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
8	7	anbi2i 675	. . . . . 6 ⊢ ((x ∈ B ∧ (x ∈ B → x ∈ A)) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
9	5, 6, 8	3bitri 262	. . . . 5 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ ¬ (x ∈ B ∧ ¬ x ∈ A)))
10	4, 9	bitr4i 243	. . . 4 ⊢ ((x ∈ B ∧ ¬ x ∈ (B ∖ A)) ↔ x ∈ (B ∩ A))
11	10	difeqri 3388	. . 3 ⊢ (B ∖ (B ∖ A)) = (B ∩ A)
12	11	eqeq1i 2360	. 2 ⊢ ((B ∖ (B ∖ A)) = A ↔ (B ∩ A) = A)
13	1, 12	bitr4i 243	1 ⊢ (A ⊆ B ↔ (B ∖ (B ∖ A)) = A)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by:  dfin4  3496