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Theorem ssequn1 4179
Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssequn1 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)

Proof of Theorem ssequn1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bicom 221 . . . 4 ((𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
2 pm4.72 949 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)))
3 elun 4147 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43bibi1i 339 . . . 4 ((𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
51, 2, 43bitr4i 303 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
65albii 1822 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
7 dfss2 3967 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfcleq 2726 . 2 ((𝐴𝐵) = 𝐵 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
96, 7, 83bitr4i 303 1 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 846  wal 1540   = wceq 1542  wcel 2107  cun 3945  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3952  df-in 3954  df-ss 3964
This theorem is referenced by:  ssequn2  4182  undif  4480  uniop  5514  pwssun  5570  cnvimassrndm  6148  unisucg  6439  ordssun  6463  ordequn  6464  onunel  6466  onun2  6469  funiunfv  7242  sorpssun  7715  ordunpr  7809  onuninsuci  7824  omun  7873  sucdom2OLD  9078  domss2  9132  findcard2s  9161  sucdom2  9202  rankopb  9843  ranksuc  9856  kmlem11  10151  fin1a2lem10  10400  trclublem  14938  trclubi  14939  trclub  14941  reltrclfv  14960  modfsummods  15735  cvgcmpce  15760  mreexexlem3d  17586  dprd2da  19904  dpjcntz  19914  dpjdisj  19915  dpjlsm  19916  dpjidcl  19920  ablfac1eu  19935  perfcls  22851  dfconn2  22905  comppfsc  23018  llycmpkgen2  23036  trfil2  23373  fixufil  23408  tsmsres  23630  ustssco  23701  ustuqtop1  23728  xrge0gsumle  24331  volsup  25055  mbfss  25145  itg2cnlem2  25262  iblss2  25305  vieta1lem2  25806  amgm  26475  wilthlem2  26553  ftalem3  26559  rpvmasum2  26995  noetalem1  27224  madeoldsuc  27359  iuninc  31770  pmtrcnel  32228  pmtrcnelor  32230  hgt750lemb  33606  rankaltopb  34889  hfun  35088  bj-prmoore  35934  nacsfix  41383  cantnfresb  42007  omabs2  42015  onsucunipr  42055  oaun2  42064  oaun3  42065  fvnonrel  42281  rclexi  42299  rtrclex  42301  trclubgNEW  42302  trclubNEW  42303  dfrtrcl5  42313  trrelsuperrel2dg  42355  iunrelexp0  42386  corcltrcl  42423  isotone1  42732  aacllem  47750
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