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Theorem uniopel 4403
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 4402 . . 3  |-  U. <. A ,  B >.  =  { A ,  B }
41, 2opi2 4374 . . 3  |-  { A ,  B }  e.  <. A ,  B >.
53, 4eqeltri 2459 . 2  |-  U. <. A ,  B >.  e.  <. A ,  B >.
6 elssuni 3987 . . 3  |-  ( <. A ,  B >.  e.  C  ->  <. A ,  B >.  C_  U. C )
76sseld 3292 . 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 set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2901   {cpr 3760   <.cop 3762   U.cuni 3959
This theorem is referenced by:  dmrnssfld  5071  unielrel  5336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rex 2657  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960
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