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Theorem ssiun2s 3796
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 2235 . 2  |-  F/_ x C
2 nfcv 2235 . . 3  |-  F/_ x D
3 nfiu1 3782 . . 3  |-  F/_ x U_ x  e.  A  B
42, 3nfss 3032 . 2  |-  F/ x  D  C_  U_ x  e.  A  B
5 ssiun2s.1 . . 3  |-  ( x  =  C  ->  B  =  D )
65sseq1d 3068 . 2  |-  ( x  =  C  ->  ( B  C_  U_ x  e.  A  B  <->  D  C_  U_ x  e.  A  B )
)
7 ssiun2 3795 . 2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
81, 4, 6, 7vtoclgaf 2698 1  |-  ( C  e.  A  ->  D  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445    C_ wss 3013   U_ciun 3752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-iun 3754
This theorem is referenced by: (None)
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