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Theorem opeluu 4339
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 3597 . . 3  |-  A  e. 
{ A ,  B }
3 opeluu.2 . . . . 5  |-  B  e. 
_V
41, 3opi2 4123 . . . 4  |-  { A ,  B }  e.  <. A ,  B >.
5 elunii 3709 . . . 4  |-  ( ( { A ,  B }  e.  <. A ,  B >.  /\  <. A ,  B >.  e.  C )  ->  { A ,  B }  e.  U. C
)
64, 5mpan 418 . . 3  |-  ( <. A ,  B >.  e.  C  ->  { A ,  B }  e.  U. C )
7 elunii 3709 . . 3  |-  ( ( A  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  A  e.  U. U. C
)
82, 6, 7sylancr 408 . 2  |-  ( <. A ,  B >.  e.  C  ->  A  e.  U.
U. C )
93prid2 3598 . . 3  |-  B  e. 
{ A ,  B }
10 elunii 3709 . . 3  |-  ( ( B  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  B  e.  U. U. C
)
119, 6, 10sylancr 408 . 2  |-  ( <. A ,  B >.  e.  C  ->  B  e.  U.
U. C )
128, 11jca 302 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 103    e. wcel 1463   _Vcvv 2658   {cpr 3496   <.cop 3498   U.cuni 3704
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705
This theorem is referenced by:  asymref  4892
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