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Theorem ssiun2s 4069
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 2516 . 2  |-  F/_ x C
2 nfcv 2516 . . 3  |-  F/_ x D
3 nfiu1 4056 . . 3  |-  F/_ x U_ x  e.  A  B
42, 3nfss 3277 . 2  |-  F/ x  D  C_  U_ x  e.  A  B
5 ssiun2s.1 . . 3  |-  ( x  =  C  ->  B  =  D )
65sseq1d 3311 . 2  |-  ( x  =  C  ->  ( B  C_  U_ x  e.  A  B  <->  D  C_  U_ x  e.  A  B )
)
7 ssiun2 4068 . 2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
81, 4, 6, 7vtoclgaf 2952 1  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3256   U_ciun 4028
This theorem is referenced by:  onfununi  6532  oaordi  6718  omordi  6738  dffi3  7364  alephordi  7881  domtriomlem  8248  pwxpndom2  8466  wunex2  8539  imasaddvallem  13674  imasvscaval  13683  iundisj2  19303  voliunlem1  19304  volsup  19310  iundisj2fi  23984  cvmliftlem10  24753  cvmliftlem13  24755  sstotbnd2  26167  bnj906  28632  bnj1137  28695  bnj1408  28736  mapdrvallem3  31812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894  df-in 3263  df-ss 3270  df-iun 4030
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