MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniop Structured version   Unicode version

Theorem uniop 4759
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 4218 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43unieqi 4260 . 2  |-  U. <. A ,  B >.  =  U. { { A } ,  { A ,  B } }
5 snex 4697 . . 3  |-  { A }  e.  _V
6 prex 4698 . . 3  |-  { A ,  B }  e.  _V
75, 6unipr 4264 . 2  |-  U. { { A } ,  { A ,  B } }  =  ( { A }  u.  { A ,  B } )
8 snsspr1 4181 . . 3  |-  { A }  C_  { A ,  B }
9 ssequn1 3670 . . 3  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 208 . 2  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
114, 7, 103eqtri 2490 1  |-  U. <. A ,  B >.  =  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   {csn 4032   {cpr 4034   <.cop 4038   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252
This theorem is referenced by:  uniopel  4760  elvvuni  5069  dmrnssfld  5271  dffv2  5946  rankxplim  8314
  Copyright terms: Public domain W3C validator