Theorem dfss4 4219

Description: Subclass defined in terms of class difference. See comments under dfun2 4220. (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 4173	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eldif 3912	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	2	notbii 322	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	anbi2i 632	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
5		elin 3918	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6		abai 836	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
7		iman 405	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8	7	anbi2i 632	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
9	5, 6, 8	3bitri 299	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
10	4, 9	bitr4i 280	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴))
11	10	difeqri 4080	. . 3 ⊢ (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴)
12	11	eqeq1i 2766	. 2 ⊢ ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴)
13	1, 12	bitr4i 280	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919

This theorem is referenced by:  ssdifim  4223  dfin4  4228  sscon34b  4254  sorpsscmpl  7712  sbthlem3  9055  fin23lem7  10267  fin23lem11  10268  compsscnvlem  10321  compssiso  10325  isf34lem4  10328  efgmnvl  19745  frlmlbs  21837  isopn2  23080  iincld  23087  iuncld  23093  clsval2  23098  ntrval2  23099  ntrdif  23100  clsdif  23101  cmclsopn  23110  opncldf1  23132  indiscld  23139  mretopd  23140  restcld  23220  pnrmopn  23391  conndisj  23464  hausllycmp  23542  kqcldsat  23781  filufint  23968  cfinufil  23976  ufilen  23978  alexsublem  24092  bcth3  25381  inmbl  25592  iccmbl  25616  mbfimaicc  25681  i1fd  25731  itgss3  25865  difuncomp  32713  iundifdifd  32721  iundifdif  32722  supppreima  32854  pmtrcnelor  33232  evlextv  33800  ist0cld  34091  cldssbrsiga  34445  unelcarsg  34570  kur14lem4  35520  cldbnd  36647  clsun  36649  mblfinlem3  38119  mblfinlem4  38120  ismblfin  38121  itg2addnclem  38131  fdc  38205  dssmapnvod  44557  ntrclsfveq1  44597  ntrclsfveq  44599  ntrclsneine0lem  44601  ntrclsiso  44604  ntrclsk2  44605  ntrclskb  44606  ntrclsk3  44607  ntrclsk13  44608  ntrclsk4  44609  clsneiel2  44646  neicvgel2  44657  salincl  46859  salexct  46869  ovnsubadd2lem  47180  lincext2  49038  opncldeqv  49484