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Theorem uniop 4268
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 3796 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43unieqi 3838 . 2  |-  U. <. A ,  B >.  =  U. { { A } ,  { A ,  B } }
5 snex 4215 . . 3  |-  { A }  e.  _V
6 prex 4216 . . 3  |-  { A ,  B }  e.  _V
75, 6unipr 3842 . 2  |-  U. { { A } ,  { A ,  B } }  =  ( { A }  u.  { A ,  B } )
8 snsspr1 3765 . . 3  |-  { A }  C_  { A ,  B }
9 ssequn1 3346 . . 3  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 201 . 2  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
114, 7, 103eqtri 2308 1  |-  U. <. A ,  B >.  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1628    e. wcel 1688   _Vcvv 2789    u. cun 3151    C_ wss 3153   {csn 3641   {cpr 3642   <.cop 3644   U.cuni 3828
This theorem is referenced by:  uniopel  4269  elvvuni  4749  dmrnssfld  4937  dffv2  5553  rankxplim  7544
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rex 2550  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829
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