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Theorem uniop 4736
Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1  |-  A  e. 
_V
opthw.2  |-  B  e. 
_V
Assertion
Ref Expression
uniop  |-  U. <. A ,  B >.  =  { A ,  B }

Proof of Theorem uniop
StepHypRef Expression
1 opthw.1 . . . 4  |-  A  e. 
_V
2 opthw.2 . . . 4  |-  B  e. 
_V
31, 2dfop 4197 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43unieqi 4239 . 2  |-  U. <. A ,  B >.  =  U. { { A } ,  { A ,  B } }
5 snex 4674 . . 3  |-  { A }  e.  _V
6 prex 4675 . . 3  |-  { A ,  B }  e.  _V
75, 6unipr 4243 . 2  |-  U. { { A } ,  { A ,  B } }  =  ( { A }  u.  { A ,  B } )
8 snsspr1 4160 . . 3  |-  { A }  C_  { A ,  B }
9 ssequn1 3656 . . 3  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 208 . 2  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
114, 7, 103eqtri 2474 1  |-  U. <. A ,  B >.  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1381    e. wcel 1802   _Vcvv 3093    u. cun 3456    C_ wss 3458   {csn 4010   {cpr 4012   <.cop 4016   U.cuni 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rex 2797  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231
This theorem is referenced by:  uniopel  4737  elvvuni  5046  dmrnssfld  5247  dffv2  5927  rankxplim  8295
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