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Theorem opeluu 4202
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 3500 . . 3  |-  A  e. 
{ A ,  B }
3 opeluu.2 . . . . 5  |-  B  e. 
_V
41, 3opi2 3990 . . . 4  |-  { A ,  B }  e.  <. A ,  B >.
5 elunii 3608 . . . 4  |-  ( ( { A ,  B }  e.  <. A ,  B >.  /\  <. A ,  B >.  e.  C )  ->  { A ,  B }  e.  U. C
)
64, 5mpan 415 . . 3  |-  ( <. A ,  B >.  e.  C  ->  { A ,  B }  e.  U. C )
7 elunii 3608 . . 3  |-  ( ( A  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  A  e.  U. U. C
)
82, 6, 7sylancr 405 . 2  |-  ( <. A ,  B >.  e.  C  ->  A  e.  U.
U. C )
93prid2 3501 . . 3  |-  B  e. 
{ A ,  B }
10 elunii 3608 . . 3  |-  ( ( B  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  B  e.  U. U. C
)
119, 6, 10sylancr 405 . 2  |-  ( <. A ,  B >.  e.  C  ->  B  e.  U.
U. C )
128, 11jca 300 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 102    e. wcel 1434   _Vcvv 2602   {cpr 3401   <.cop 3403   U.cuni 3603
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  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-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604
This theorem is referenced by:  asymref  4734
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