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Theorem ssequn1 3517
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  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )

Proof of Theorem ssequn1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 bicom 192 . . . 4  |-  ( ( x  e.  B  <->  ( x  e.  A  \/  x  e.  B ) )  <->  ( (
x  e.  A  \/  x  e.  B )  <->  x  e.  B ) )
2 pm4.72 847 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  B  <->  ( x  e.  A  \/  x  e.  B ) ) )
3 elun 3488 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
43bibi1i 306 . . . 4  |-  ( ( x  e.  ( A  u.  B )  <->  x  e.  B )  <->  ( (
x  e.  A  \/  x  e.  B )  <->  x  e.  B ) )
51, 2, 43bitr4i 269 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  ( A  u.  B )  <->  x  e.  B ) )
65albii 1575 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  ( A  u.  B )  <-> 
x  e.  B ) )
7 dfss2 3337 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
8 dfcleq 2430 . 2  |-  ( ( A  u.  B )  =  B  <->  A. x
( x  e.  ( A  u.  B )  <-> 
x  e.  B ) )
96, 7, 83bitr4i 269 1  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358   A.wal 1549    = wceq 1652    e. wcel 1725    u. cun 3318    C_ wss 3320
This theorem is referenced by:  ssequn2  3520  undif  3708  uniop  4459  pwssun  4489  unisuc  4657  ordssun  4681  ordequn  4682  onun2i  4697  ordunpr  4806  onuninsuci  4820  funiunfv  5995  sorpssun  6529  domss2  7266  sucdom2  7303  findcard2s  7349  rankopb  7778  ranksuc  7791  kmlem11  8040  fin1a2lem10  8289  cvgcmpce  12597  mreexexlem3d  13871  dprd2da  15600  dpjcntz  15610  dpjdisj  15611  dpjlsm  15612  dpjidcl  15616  ablfac1eu  15631  perfcls  17429  dfcon2  17482  llycmpkgen2  17582  trfil2  17919  fixufil  17954  tsmsres  18173  ustssco  18244  ustuqtop1  18271  xrge0gsumle  18864  volsup  19450  mbfss  19538  itg2cnlem2  19654  iblss2  19697  vieta1lem2  20228  amgm  20829  wilthlem2  20852  ftalem3  20857  rpvmasum2  21206  iuninc  24011  rankaltopb  25824  hfun  26119  comppfsc  26387  nacsfix  26766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334
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