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Theorem elunii 3736
Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)
Assertion
Ref Expression
elunii  |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C
)

Proof of Theorem elunii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2201 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eleq1 2200 . . . . 5  |-  ( x  =  B  ->  (
x  e.  C  <->  B  e.  C ) )
31, 2anbi12d 464 . . . 4  |-  ( x  =  B  ->  (
( A  e.  x  /\  x  e.  C
)  <->  ( A  e.  B  /\  B  e.  C ) ) )
43spcegv 2769 . . 3  |-  ( B  e.  C  ->  (
( A  e.  B  /\  B  e.  C
)  ->  E. x
( A  e.  x  /\  x  e.  C
) ) )
54anabsi7 570 . 2  |-  ( ( A  e.  B  /\  B  e.  C )  ->  E. x ( A  e.  x  /\  x  e.  C ) )
6 eluni 3734 . 2  |-  ( A  e.  U. C  <->  E. x
( A  e.  x  /\  x  e.  C
) )
75, 6sylibr 133 1  |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   U.cuni 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-uni 3732
This theorem is referenced by:  ssuni  3753  unipw  4134  opeluu  4366  sucunielr  4421  unon  4422  ordunisuc2r  4425  tfrlemibxssdm  6217  tfr1onlemsucaccv  6231  tfr1onlembxssdm  6233  tfrcllemsucaccv  6244  tfrcllembxssdm  6246  tgss2  12237  neipsm  12312  unirnblps  12580  unirnbl  12581  blbas  12591
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