MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssequn1 Structured version   Visualization version   GIF version

Theorem ssequn1 3745
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 211 . . . 4 ((𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
2 pm4.72 916 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)))
3 elun 3715 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43bibi1i 327 . . . 4 ((𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
51, 2, 43bitr4i 291 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
65albii 1737 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
7 dfss2 3557 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfcleq 2604 . 2 ((𝐴𝐵) = 𝐵 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
96, 7, 83bitr4i 291 1 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wal 1473   = wceq 1475  wcel 1977  cun 3538  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-un 3545  df-in 3547  df-ss 3554
This theorem is referenced by:  ssequn2  3748  undif  4001  uniop  4893  pwssun  4934  unisuc  5704  ordssun  5730  ordequn  5731  onun2i  5746  funiunfv  6388  sorpssun  6820  ordunpr  6896  onuninsuci  6910  domss2  7982  sucdom2  8019  findcard2s  8064  rankopb  8576  ranksuc  8589  kmlem11  8843  fin1a2lem10  9092  trclublem  13531  trclubi  13532  trclubiOLD  13533  trclub  13536  reltrclfv  13555  modfsummods  14315  cvgcmpce  14340  mreexexlem3d  16078  dprd2da  18213  dpjcntz  18223  dpjdisj  18224  dpjlsm  18225  dpjidcl  18229  ablfac1eu  18244  perfcls  20927  dfcon2  20980  comppfsc  21093  llycmpkgen2  21111  trfil2  21449  fixufil  21484  tsmsres  21705  ustssco  21776  ustuqtop1  21803  xrge0gsumle  22392  volsup  23076  mbfss  23164  itg2cnlem2  23280  iblss2  23323  vieta1lem2  23815  amgm  24462  wilthlem2  24540  ftalem3  24546  rpvmasum2  24946  iuninc  28555  rankaltopb  31050  hfun  31249  nacsfix  36087  fvnonrel  36716  rclexi  36735  rtrclex  36737  trclubgNEW  36738  trclubNEW  36739  dfrtrcl5  36749  trrelsuperrel2dg  36776  iunrelexp0  36807  corcltrcl  36844  isotone1  37160  aacllem  42309
  Copyright terms: Public domain W3C validator