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Theorem uniopel 4286
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 4285 . . 3  |-  U. <. A ,  B >.  =  { A ,  B }
41, 2opi2 4257 . . 3  |-  { A ,  B }  e.  <. A ,  B >.
53, 4eqeltri 2366 . 2  |-  U. <. A ,  B >.  e.  <. A ,  B >.
6 elssuni 3871 . . 3  |-  ( <. A ,  B >.  e.  C  ->  <. A ,  B >.  C_  U. C )
76sseld 3192 . 2  |-  ( <. A ,  B >.  e.  C  ->  ( U. <. A ,  B >.  e. 
<. A ,  B >.  ->  U. <. A ,  B >.  e.  U. C ) )
85, 7mpi 16 1  |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801   {cpr 3654   <.cop 3656   U.cuni 3843
This theorem is referenced by:  dmrnssfld  4954  unielrel  5213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844
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