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Theorem ssiun2s 4337
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 2582 . 2  |-  F/_ x C
2 nfcv 2582 . . 3  |-  F/_ x D
3 nfiu1 4323 . . 3  |-  F/_ x U_ x  e.  A  B
42, 3nfss 3454 . 2  |-  F/ x  D  C_  U_ x  e.  A  B
5 ssiun2s.1 . . 3  |-  ( x  =  C  ->  B  =  D )
65sseq1d 3488 . 2  |-  ( x  =  C  ->  ( B  C_  U_ x  e.  A  B  <->  D  C_  U_ x  e.  A  B )
)
7 ssiun2 4336 . 2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
81, 4, 6, 7vtoclgaf 3141 1  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867    C_ wss 3433   U_ciun 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-v 3080  df-in 3440  df-ss 3447  df-iun 4295
This theorem is referenced by:  onfununi  7059  oaordi  7246  omordi  7266  dffi3  7942  alephordi  8494  domtriomlem  8861  pwxpndom2  9079  wunex2  9152  imasaddvallem  15379  imasvscaval  15388  iundisj2  22376  voliunlem1  22377  volsup  22383  iundisj2fi  28206  bnj906  29526  bnj1137  29589  bnj1408  29630  cvmliftlem10  29802  cvmliftlem13  29804  sstotbnd2  31810  mapdrvallem3  34923  fvmptiunrelexplb0d  35919  fvmptiunrelexplb1d  35921  corclrcl  35942  trclrelexplem  35946  corcltrcl  35974  cotrclrcl  35977  iundjiunlem  37810  caratheodorylem1  37860
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