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Theorem uniopel 4760
Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1  |-  A  e. 
_V
opthw.2  |-  B  e. 
_V
Assertion
Ref Expression
uniopel  |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C
)

Proof of Theorem uniopel
StepHypRef Expression
1 opthw.1 . . . 4  |-  A  e. 
_V
2 opthw.2 . . . 4  |-  B  e. 
_V
31, 2uniop 4759 . . 3  |-  U. <. A ,  B >.  =  { A ,  B }
41, 2opi2 4724 . . 3  |-  { A ,  B }  e.  <. A ,  B >.
53, 4eqeltri 2541 . 2  |-  U. <. A ,  B >.  e.  <. A ,  B >.
6 elssuni 4281 . . 3  |-  ( <. A ,  B >.  e.  C  ->  <. A ,  B >.  C_  U. C )
76sseld 3498 . 2  |-  ( <. A ,  B >.  e.  C  ->  ( U. <. A ,  B >.  e. 
<. A ,  B >.  ->  U. <. A ,  B >.  e.  U. C ) )
85, 7mpi 17 1  |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819   _Vcvv 3109   {cpr 4034   <.cop 4038   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252
This theorem is referenced by:  dmrnssfld  5271  unielrel  5538
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