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Theorem dfss4 3820
 Description: Subclass defined in terms of class difference. See comments under dfun2 3821. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfss4 (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴)

Proof of Theorem dfss4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseqin2 3779 . 2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
2 eldif 3550 . . . . . . 7 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
32notbii 309 . . . . . 6 𝑥 ∈ (𝐵𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
43anbi2i 726 . . . . 5 ((𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
5 elin 3758 . . . . . 6 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
6 abai 832 . . . . . 6 ((𝑥𝐵𝑥𝐴) ↔ (𝑥𝐵 ∧ (𝑥𝐵𝑥𝐴)))
7 iman 439 . . . . . . 7 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
87anbi2i 726 . . . . . 6 ((𝑥𝐵 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
95, 6, 83bitri 285 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
104, 9bitr4i 266 . . . 4 ((𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥 ∈ (𝐵𝐴))
1110difeqri 3692 . . 3 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
1211eqeq1i 2615 . 2 ((𝐵 ∖ (𝐵𝐴)) = 𝐴 ↔ (𝐵𝐴) = 𝐴)
131, 12bitr4i 266 1 (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554 This theorem is referenced by:  dfin4  3826  sorpsscmpl  6846  sbthlem3  7957  fin23lem7  9021  fin23lem11  9022  compsscnvlem  9075  compssiso  9079  isf34lem4  9082  efgmnvl  17950  frlmlbs  19955  isopn2  20646  iincld  20653  iuncld  20659  clsval2  20664  ntrval2  20665  ntrdif  20666  clsdif  20667  cmclsopn  20676  opncldf1  20698  indiscld  20705  mretopd  20706  restcld  20786  pnrmopn  20957  conndisj  21029  hausllycmp  21107  kqcldsat  21346  filufint  21534  cfinufil  21542  ufilen  21544  alexsublem  21658  bcth3  22936  inmbl  23117  iccmbl  23141  mbfimaicc  23206  i1fd  23254  itgss3  23387  frgrawopreg2  26578  difuncomp  28752  iundifdifd  28762  iundifdif  28763  cldssbrsiga  29577  unelcarsg  29701  kur14lem4  30445  cldbnd  31491  clsun  31493  mblfinlem3  32618  mblfinlem4  32619  ismblfin  32620  itg2addnclem  32631  fdc  32711  dssmapnvod  37334  sscon34b  37337  ntrclsfveq1  37378  ntrclsfveq  37380  ntrclsneine0lem  37382  ntrclsiso  37385  ntrclsk2  37386  ntrclskb  37387  ntrclsk3  37388  ntrclsk13  37389  ntrclsk4  37390  clsneiel2  37427  neicvgel2  37438  salincl  39219  salexct  39228  ovnsubadd2lem  39535  frgrwopreg2  41488  lincext2  42038
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