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Theorem opeluu 4463
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 3675 . . 3  |-  A  e. 
{ A ,  B }
3 opeluu.2 . . . . 5  |-  B  e. 
_V
41, 3opi2 4178 . . . 4  |-  { A ,  B }  e.  <. A ,  B >.
5 elunii 3773 . . . 4  |-  ( ( { A ,  B }  e.  <. A ,  B >.  /\  <. A ,  B >.  e.  C )  ->  { A ,  B }  e.  U. C
)
64, 5mpan 654 . . 3  |-  ( <. A ,  B >.  e.  C  ->  { A ,  B }  e.  U. C )
7 elunii 3773 . . 3  |-  ( ( A  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  A  e.  U. U. C
)
82, 6, 7sylancr 647 . 2  |-  ( <. A ,  B >.  e.  C  ->  A  e.  U.
U. C )
93prid2 3676 . . 3  |-  B  e. 
{ A ,  B }
10 elunii 3773 . . 3  |-  ( ( B  e.  { A ,  B }  /\  { A ,  B }  e.  U. C )  ->  B  e.  U. U. C
)
119, 6, 10sylancr 647 . 2  |-  ( <. A ,  B >.  e.  C  ->  B  e.  U.
U. C )
128, 11jca 520 1  |-  ( <. A ,  B >.  e.  C  ->  ( A  e.  U. U. C  /\  B  e.  U. U. C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   _Vcvv 2740   {cpr 3582   <.cop 3584   U.cuni 3768
This theorem is referenced by:  asymref  5012  asymref2  5013  wrdexb  11379  dfdir2  24623
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769
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