Theorem dfss4 4217

Description: Subclass defined in terms of class difference. See comments under dfun2 4218. (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 4123	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eldif 4052	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	2	notbii 323	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	anbi2i 635	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
5		elin 4096	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6		abai 839	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
7		iman 407	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8	7	anbi2i 635	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
9	5, 6, 8	3bitri 300	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
10	4, 9	bitr4i 281	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴))
11	10	difeqri 4068	. . 3 ⊢ (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴)
12	11	eqeq1i 2768	. 2 ⊢ ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴)
13	1, 12	bitr4i 281	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094

This theorem is used by:  ssdifim  4221  dfin4  4226  sscon34b  4252  sorpsscmpl  7780  sbthlem3  9144  fin23lem7  10431  fin23lem11  10432  compsscnvlem  10485  compssiso  10489  isf34lem4  10492  efgmnvl  19970  frlmlbs  22147  isopn2  23394  iincld  23401  iuncld  23407  clsval2  23412  ntrval2  23413  ntrdif  23414  clsdif  23415  cmclsopn  23424  opncldf1  23446  indiscld  23453  mretopd  23454  restcld  23534  pnrmopn  23705  conndisj  23778  hausllycmp  23857  kqcldsat  24096  filufint  24283  cfinufil  24291  ufilen  24293  alexsublem  24407  bcth3  25696  inmbl  25907  iccmbl  25931  mbfimaicc  25996  i1fd  26046  itgss3  26179  difuncomp  33236  iundifdifd  33244  iundifdif  33245  supppreima  33371  pmtrcnelor  33741  evlextv  34278  ist0cld  34569  difelsiga  34871  cldssbrsiga  34924  unelcarsg  35048  kur14lem4  36175  cldbnd  37322  clsun  37324  mblfinlem3  38838  mblfinlem4  38839  ismblfin  38840  itg2addnclem  38850  fdc  38938  dssmapnvod  45275  ntrclsfveq1  45315  ntrclsfveq  45317  ntrclsneine0lem  45319  ntrclsiso  45322  ntrclsk2  45323  ntrclskb  45324  ntrclsk3  45325  ntrclsk13  45326  ntrclsk4  45327  clsneiel2  45364  neicvgel2  45375  salincl  47601  salexct  47611  ovnsubadd2lem  47922  lincext2  49892  opncldbid  50341