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Theorem opeluu 4526
Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
opeluu.1  |-  A  e. 
_V
opeluu.2  |-  B  e. 
_V
Assertion
Ref Expression
opeluu  |-  ( <. A ,  B >.  e.  C  ->  ( A  e.  U. U. C  /\  B  e.  U. U. C
) )

Proof of Theorem opeluu
StepHypRef Expression
1 opeluu.1 . . . 4  |-  A  e. 
_V
21prid1 3734 . . 3  |-  A  e. 
{ A ,  B }
3 opeluu.2 . . . . 5  |-  B  e. 
_V
41, 3opi2 4241 . . . 4  |-  { A ,  B }  e.  <. A ,  B >.
5 elunii 3832 . . . 4  |-  ( ( { A ,  B }  e.  <. A ,  B >.  /\  <. A ,  B >.  e.  C )  ->  { A ,  B }  e.  U. C
)
64, 5mpan 651 . . 3  |-  ( <. A ,  B >.  e.  C  ->  { A ,  B }  e.  U. C )
7 elunii 3832 . . 3  |-  ( ( A  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  A  e.  U. U. C
)
82, 6, 7sylancr 644 . 2  |-  ( <. A ,  B >.  e.  C  ->  A  e.  U.
U. C )
93prid2 3735 . . 3  |-  B  e. 
{ A ,  B }
10 elunii 3832 . . 3  |-  ( ( B  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  B  e.  U. U. C
)
119, 6, 10sylancr 644 . 2  |-  ( <. A ,  B >.  e.  C  ->  B  e.  U.
U. C )
128, 11jca 518 1  |-  ( <. A ,  B >.  e.  C  ->  ( A  e.  U. U. C  /\  B  e.  U. U. C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788   {cpr 3641   <.cop 3643   U.cuni 3827
This theorem is referenced by:  asymref  5059  asymref2  5060  wrdexb  11449  dfdir2  25291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828
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