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Theorem soex 6728
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 461 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  =  (/) )
2 0ex 4577 . . 3  |-  (/)  e.  _V
31, 2syl6eqel 2563 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  e.  _V )
4 n0 3794 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 snex 4688 . . . . . . . . 9  |-  { x }  e.  _V
6 dmexg 6716 . . . . . . . . . 10  |-  ( R  e.  V  ->  dom  R  e.  _V )
7 rnexg 6717 . . . . . . . . . 10  |-  ( R  e.  V  ->  ran  R  e.  _V )
8 unexg 6586 . . . . . . . . . 10  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
96, 7, 8syl2anc 661 . . . . . . . . 9  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
10 unexg 6586 . . . . . . . . 9  |-  ( ( { x }  e.  _V  /\  ( dom  R  u.  ran  R )  e. 
_V )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
115, 9, 10sylancr 663 . . . . . . . 8  |-  ( R  e.  V  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
1211ad2antlr 726 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( { x }  u.  ( dom  R  u.  ran  R ) )  e.  _V )
13 sossfld 5454 . . . . . . . . 9  |-  ( ( R  Or  A  /\  x  e.  A )  ->  ( A  \  {
x } )  C_  ( dom  R  u.  ran  R ) )
1413adantlr 714 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  ( A  \  { x }
)  C_  ( dom  R  u.  ran  R ) )
15 ssundif 3910 . . . . . . . 8  |-  ( A 
C_  ( { x }  u.  ( dom  R  u.  ran  R ) )  <->  ( A  \  { x } ) 
C_  ( dom  R  u.  ran  R ) )
1614, 15sylibr 212 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  C_  ( { x }  u.  ( dom  R  u.  ran  R ) ) )
1712, 16ssexd 4594 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  x  e.  A )  ->  A  e.  _V )
1817ex 434 . . . . 5  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( x  e.  A  ->  A  e.  _V )
)
1918exlimdv 1700 . . . 4  |-  ( ( R  Or  A  /\  R  e.  V )  ->  ( E. x  x  e.  A  ->  A  e.  _V ) )
2019imp 429 . . 3  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  E. x  x  e.  A )  ->  A  e.  _V )
214, 20sylan2b 475 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =/=  (/) )  ->  A  e.  _V )
223, 21pm2.61dane 2785 1  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027    Or wor 4799   dom cdm 4999   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  ween  8417  zorn2lem1  8877  zorn2lem4  8880
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