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

Theorem soex 6742
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 4587 . . 3  |-  (/)  e.  _V
31, 2syl6eqel 2553 . 2  |-  ( ( ( R  Or  A  /\  R  e.  V
)  /\  A  =  (/) )  ->  A  e.  _V )
4 n0 3803 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
5 snex 4697 . . . . . . . . 9  |-  { x }  e.  _V
6 dmexg 6730 . . . . . . . . . 10  |-  ( R  e.  V  ->  dom  R  e.  _V )
7 rnexg 6731 . . . . . . . . . 10  |-  ( R  e.  V  ->  ran  R  e.  _V )
8 unexg 6600 . . . . . . . . . 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 6600 . . . . . . . . 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 5460 . . . . . . . . 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 3914 . . . . . . . 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 4603 . . . . . 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 1725 . . . 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 2775 1  |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3793   {csn 4032    Or wor 4808   dom cdm 5008   ran crn 5009
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-8 1821  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  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  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  df-br 4457  df-opab 4516  df-po 4809  df-so 4810  df-cnv 5016  df-dm 5018  df-rn 5019
This theorem is referenced by:  ween  8433  zorn2lem1  8893  zorn2lem4  8896
  Copyright terms: Public domain W3C validator