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Theorem uniop 4451
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 3975 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43unieqi 4017 . 2  |-  U. <. A ,  B >.  =  U. { { A } ,  { A ,  B } }
5 snex 4397 . . 3  |-  { A }  e.  _V
6 prex 4398 . . 3  |-  { A ,  B }  e.  _V
75, 6unipr 4021 . 2  |-  U. { { A } ,  { A ,  B } }  =  ( { A }  u.  { A ,  B } )
8 snsspr1 3939 . . 3  |-  { A }  C_  { A ,  B }
9 ssequn1 3509 . . 3  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 200 . 2  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
114, 7, 103eqtri 2459 1  |-  U. <. A ,  B >.  =  { A ,  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    C_ wss 3312   {csn 3806   {cpr 3807   <.cop 3809   U.cuni 4007
This theorem is referenced by:  uniopel  4452  elvvuni  4929  dmrnssfld  5120  dffv2  5787  rankxplim  7792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008
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