Theorem soex 7109

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.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ( R Or A /\ R e. V ) /\ A = (/) ) -> A = (/) )
2		0ex 4790	. . 3 |- (/) e. _V
3	1, 2	syl6eqel 2709	. 2 |- ( ( ( R Or A /\ R e. V ) /\ A = (/) ) -> A e. _V )
4		n0 3931	. . 3 |- ( A =/= (/) <-> E. x x e. A )
5		snex 4908	. . . . . . . . 9 |- { x } e. _V
6		dmexg 7097	. . . . . . . . . 10 |- ( R e. V -> dom R e. _V )
7		rnexg 7098	. . . . . . . . . 10 |- ( R e. V -> ran R e. _V )
8		unexg 6959	. . . . . . . . . 10 |- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R u. ran R ) e. _V )
9	6, 7, 8	syl2anc 693	. . . . . . . . 9 |- ( R e. V -> ( dom R u. ran R ) e. _V )
10		unexg 6959	. . . . . . . . 9 |- ( ( { x } e. _V /\ ( dom R u. ran R ) e. _V ) -> ( { x } u. ( dom R u. ran R ) ) e. _V )
11	5, 9, 10	sylancr 695	. . . . . . . 8 |- ( R e. V -> ( { x } u. ( dom R u. ran R ) ) e. _V )
12	11	ad2antlr 763	. . . . . . 7 |- ( ( ( R Or A /\ R e. V ) /\ x e. A ) -> ( { x } u. ( dom R u. ran R ) ) e. _V )
13		sossfld 5580	. . . . . . . . 9 |- ( ( R Or A /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
14	13	adantlr 751	. . . . . . . 8 |- ( ( ( R Or A /\ R e. V ) /\ x e. A ) -> ( A \ { x } ) C_ ( dom R u. ran R ) )
15		ssundif 4052	. . . . . . . 8 |- ( A C_ ( { x } u. ( dom R u. ran R ) ) <-> ( A \ { x } ) C_ ( dom R u. ran R ) )
16	14, 15	sylibr 224	. . . . . . 7 |- ( ( ( R Or A /\ R e. V ) /\ x e. A ) -> A C_ ( { x } u. ( dom R u. ran R ) ) )
17	12, 16	ssexd 4805	. . . . . 6 |- ( ( ( R Or A /\ R e. V ) /\ x e. A ) -> A e. _V )
18	17	ex 450	. . . . 5 |- ( ( R Or A /\ R e. V ) -> ( x e. A -> A e. _V ) )
19	18	exlimdv 1861	. . . 4 |- ( ( R Or A /\ R e. V ) -> ( E. x x e. A -> A e. _V ) )
20	19	imp 445	. . 3 |- ( ( ( R Or A /\ R e. V ) /\ E. x x e. A ) -> A e. _V )
21	4, 20	sylan2b 492	. 2 |- ( ( ( R Or A /\ R e. V ) /\ A =/= (/) ) -> A e. _V )
22	3, 21	pm2.61dane 2881	1 |- ( ( R Or A /\ R e. V ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   Or wor 5034   dom cdm 5114   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  ween  8858  zorn2lem1  9318  zorn2lem4  9321