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Theorem soex 5122
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 447 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  =  (/) )
2 0ex 4150 . . 3  |-  (/)  e.  _V
31, 2syl6eqel 2371 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  e.  _V )
4 n0 3464 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 sossfld 5120 . . . . . . . . 9  |-  ( ( R  Or  A  /\  x  e.  A )  ->  ( A  \  {
x } )  C_  ( dom  R  u.  ran  R ) )
65adantlr 695 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( A  \  { x }
)  C_  ( dom  R  u.  ran  R ) )
7 ssundif 3537 . . . . . . . 8  |-  ( A 
C_  ( { x }  u.  ( dom  R  u.  ran  R ) )  <->  ( A  \  { x } ) 
C_  ( dom  R  u.  ran  R ) )
86, 7sylibr 203 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  C_  ( { x }  u.  ( dom  R  u.  ran  R ) ) )
9 snex 4216 . . . . . . . . 9  |-  { x }  e.  _V
10 dmexg 4939 . . . . . . . . . 10  |-  ( R  e.  V  ->  dom  R  e.  _V )
11 rnexg 4940 . . . . . . . . . 10  |-  ( R  e.  V  ->  ran  R  e.  _V )
12 unexg 4521 . . . . . . . . . 10  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
1310, 11, 12syl2anc 642 . . . . . . . . 9  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
14 unexg 4521 . . . . . . . . 9  |-  ( ( { x }  e.  _V  /\  ( dom  R  u.  ran  R )  e. 
_V )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
159, 13, 14sylancr 644 . . . . . . . 8  |-  ( R  e.  V  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
1615ad2antlr 707 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
17 ssexg 4160 . . . . . . 7  |-  ( ( A  C_  ( {
x }  u.  ( dom  R  u.  ran  R
) )  /\  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )  ->  A  e.  _V )
188, 16, 17syl2anc 642 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  e.  _V )
1918ex 423 . . . . 5  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( x  e.  A  ->  A  e.  _V )
)
2019exlimdv 1664 . . . 4  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( E. x  x  e.  A  ->  A  e.  _V ) )
2120imp 418 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  E. x  x  e.  A )  ->  A  e.  _V )
224, 21sylan2b 461 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =/=  (/) )  ->  A  e.  _V )
233, 22pm2.61dane 2524 1  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640    Or wor 4313   dom cdm 4689   ran crn 4690
This theorem is referenced by:  ween  7662  zorn2lem1  8123  zorn2lem4  8126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-cnv 4697  df-dm 4699  df-rn 4700
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