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Theorem ssiun2s 3742
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 2223 . 2  |-  F/_ x C
2 nfcv 2223 . . 3  |-  F/_ x D
3 nfiu1 3728 . . 3  |-  F/_ x U_ x  e.  A  B
42, 3nfss 3001 . 2  |-  F/ x  D  C_  U_ x  e.  A  B
5 ssiun2s.1 . . 3  |-  ( x  =  C  ->  B  =  D )
65sseq1d 3035 . 2  |-  ( x  =  C  ->  ( B  C_  U_ x  e.  A  B  <->  D  C_  U_ x  e.  A  B )
)
7 ssiun2 3741 . 2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
81, 4, 6, 7vtoclgaf 2672 1  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    C_ wss 2982   U_ciun 3698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2988  df-ss 2995  df-iun 3700
This theorem is referenced by: (None)
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