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Theorem opeluu 4525
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 3735 . . 3  |-  A  e. 
{ A ,  B }
3 opeluu.2 . . . . 5  |-  B  e. 
_V
41, 3opi2 4240 . . . 4  |-  { A ,  B }  e.  <. A ,  B >.
5 elunii 3833 . . . 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 3833 . . 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 3736 . . 3  |-  B  e. 
{ A ,  B }
10 elunii 3833 . . 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 1685   _Vcvv 2789   {cpr 3642   <.cop 3644   U.cuni 3828
This theorem is referenced by:  asymref  5058  asymref2  5059  wrdexb  11445  dfdir2  24702
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829
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