MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssiun2s Unicode version

Theorem ssiun2s 3962
Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
Hypothesis
Ref Expression
ssiun2s.1  |-  ( x  =  C  ->  B  =  D )
Assertion
Ref Expression
ssiun2s  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hint:    B( x)

Proof of Theorem ssiun2s
StepHypRef Expression
1 nfcv 2432 . 2  |-  F/_ x C
2 nfcv 2432 . . 3  |-  F/_ x D
3 nfiu1 3949 . . 3  |-  F/_ x U_ x  e.  A  B
42, 3nfss 3186 . 2  |-  F/ x  D  C_  U_ x  e.  A  B
5 ssiun2s.1 . . 3  |-  ( x  =  C  ->  B  =  D )
65sseq1d 3218 . 2  |-  ( x  =  C  ->  ( B  C_  U_ x  e.  A  B  <->  D  C_  U_ x  e.  A  B )
)
7 ssiun2 3961 . 2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
81, 4, 6, 7vtoclgaf 2861 1  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165   U_ciun 3921
This theorem is referenced by:  onfununi  6374  oaordi  6560  omordi  6580  dffi3  7200  alephordi  7717  domtriomlem  8084  pwxpndom2  8303  wunex2  8376  imasaddvallem  13447  imasvscaval  13456  iundisj2  18922  voliunlem1  18923  volsup  18929  iundisj2fi  23379  cvmliftlem10  23840  cvmliftlem13  23842  sstotbnd2  26601  bnj906  29278  bnj1137  29341  bnj1408  29382  mapdrvallem3  32458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-iun 3923
  Copyright terms: Public domain W3C validator