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Theorem uniopel 4592
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 4591 . . 3  |-  U. <. A ,  B >.  =  { A ,  B }
41, 2opi2 4557 . . 3  |-  { A ,  B }  e.  <. A ,  B >.
53, 4eqeltri 2511 . 2  |-  U. <. A ,  B >.  e.  <. A ,  B >.
6 elssuni 4118 . . 3  |-  ( <. A ,  B >.  e.  C  ->  <. A ,  B >.  C_  U. C )
76sseld 3352 . 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 1761   _Vcvv 2970   {cpr 3876   <.cop 3880   U.cuni 4088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-rex 2719  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089
This theorem is referenced by:  dmrnssfld  5094  unielrel  5359
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