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Theorem elunii 3614
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 2143 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eleq1 2142 . . . . 5  |-  ( x  =  B  ->  (
x  e.  C  <->  B  e.  C ) )
31, 2anbi12d 457 . . . 4  |-  ( x  =  B  ->  (
( A  e.  x  /\  x  e.  C
)  <->  ( A  e.  B  /\  B  e.  C ) ) )
43spcegv 2687 . . 3  |-  ( B  e.  C  ->  (
( A  e.  B  /\  B  e.  C
)  ->  E. x
( A  e.  x  /\  x  e.  C
) ) )
54anabsi7 546 . 2  |-  ( ( A  e.  B  /\  B  e.  C )  ->  E. x ( A  e.  x  /\  x  e.  C ) )
6 eluni 3612 . 2  |-  ( A  e.  U. C  <->  E. x
( A  e.  x  /\  x  e.  C
) )
75, 6sylibr 132 1  |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   U.cuni 3609
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-uni 3610
This theorem is referenced by:  ssuni  3631  unipw  3980  opeluu  4208  sucunielr  4262  unon  4263  ordunisuc2r  4266  tfrlemibxssdm  5976  tfr1onlemsucaccv  5990  tfr1onlembxssdm  5992  tfrcllemsucaccv  6003  tfrcllembxssdm  6005
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