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Theorem soex 5311
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 4331 . . 3  |-  (/)  e.  _V
31, 2syl6eqel 2523 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  e.  _V )
4 n0 3629 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 snex 4397 . . . . . . . . 9  |-  { x }  e.  _V
6 dmexg 5122 . . . . . . . . . 10  |-  ( R  e.  V  ->  dom  R  e.  _V )
7 rnexg 5123 . . . . . . . . . 10  |-  ( R  e.  V  ->  ran  R  e.  _V )
8 unexg 4702 . . . . . . . . . 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 4702 . . . . . . . . 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 5309 . . . . . . . . 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 3703 . . . . . . . 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 4342 . . . . . 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 1646 . . . 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 2676 1  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806    Or wor 4494   dom cdm 4870   ran crn 4871
This theorem is referenced by:  ween  7908  zorn2lem1  8368  zorn2lem4  8371
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-13 1727  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  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  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  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-cnv 4878  df-dm 4880  df-rn 4881
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