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Theorem uniop 4743
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 4205 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43unieqi 4247 . 2  |-  U. <. A ,  B >.  =  U. { { A } ,  { A ,  B } }
5 snex 4681 . . 3  |-  { A }  e.  _V
6 prex 4682 . . 3  |-  { A ,  B }  e.  _V
75, 6unipr 4251 . 2  |-  U. { { A } ,  { A ,  B } }  =  ( { A }  u.  { A ,  B } )
8 snsspr1 4169 . . 3  |-  { A }  C_  { A ,  B }
9 ssequn1 3667 . . 3  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 208 . 2  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
114, 7, 103eqtri 2493 1  |-  U. <. A ,  B >.  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   _Vcvv 3106    u. cun 3467    C_ wss 3469   {csn 4020   {cpr 4022   <.cop 4026   U.cuni 4238
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 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rex 2813  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239
This theorem is referenced by:  uniopel  4744  elvvuni  5052  dmrnssfld  5252  dffv2  5931  rankxplim  8286
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