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Theorem soex 5261
Description: If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
soex  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )

Proof of Theorem soex
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  =  (/) )
2 0ex 4282 . . 3  |-  (/)  e.  _V
31, 2syl6eqel 2477 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  e.  _V )
4 n0 3582 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 snex 4348 . . . . . . . . 9  |-  { x }  e.  _V
6 dmexg 5072 . . . . . . . . . 10  |-  ( R  e.  V  ->  dom  R  e.  _V )
7 rnexg 5073 . . . . . . . . . 10  |-  ( R  e.  V  ->  ran  R  e.  _V )
8 unexg 4652 . . . . . . . . . 10  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
96, 7, 8syl2anc 643 . . . . . . . . 9  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
10 unexg 4652 . . . . . . . . 9  |-  ( ( { x }  e.  _V  /\  ( dom  R  u.  ran  R )  e. 
_V )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
115, 9, 10sylancr 645 . . . . . . . 8  |-  ( R  e.  V  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
1211ad2antlr 708 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
13 sossfld 5259 . . . . . . . . 9  |-  ( ( R  Or  A  /\  x  e.  A )  ->  ( A  \  {
x } )  C_  ( dom  R  u.  ran  R ) )
1413adantlr 696 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( A  \  { x }
)  C_  ( dom  R  u.  ran  R ) )
15 ssundif 3656 . . . . . . . 8  |-  ( A 
C_  ( { x }  u.  ( dom  R  u.  ran  R ) )  <->  ( A  \  { x } ) 
C_  ( dom  R  u.  ran  R ) )
1614, 15sylibr 204 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  C_  ( { x }  u.  ( dom  R  u.  ran  R ) ) )
1712, 16ssexd 4293 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  e.  _V )
1817ex 424 . . . . 5  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( x  e.  A  ->  A  e.  _V )
)
1918exlimdv 1643 . . . 4  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( E. x  x  e.  A  ->  A  e.  _V ) )
2019imp 419 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  E. x  x  e.  A )  ->  A  e.  _V )
214, 20sylan2b 462 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =/=  (/) )  ->  A  e.  _V )
223, 21pm2.61dane 2630 1  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2552   _Vcvv 2901    \ cdif 3262    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759    Or wor 4445   dom cdm 4820   ran crn 4821
This theorem is referenced by:  ween  7851  zorn2lem1  8311  zorn2lem4  8314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-po 4446  df-so 4447  df-cnv 4828  df-dm 4830  df-rn 4831
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