Theorem fin2i2 9140

Description: A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin2i2	|- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> |^| B e. B )

Proof of Theorem fin2i2

Dummy variables c m n w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . 3 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> B C_ ~P A )
2		simpll 790	. . . . 5 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> A e. FinII )
3		ssrab2 3687	. . . . . 6 |- { c e. ~P A | ( A \ c ) e. B } C_ ~P A
4	3	a1i 11	. . . . 5 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> { c e. ~P A | ( A \ c ) e. B } C_ ~P A )
5		simprl 794	. . . . . 6 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> B =/= (/) )
6		fin23lem7 9138	. . . . . 6 |- ( ( A e. FinII /\ B C_ ~P A /\ B =/= (/) ) -> { c e. ~P A | ( A \ c ) e. B } =/= (/) )
7	2, 1, 5, 6	syl3anc 1326	. . . . 5 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> { c e. ~P A | ( A \ c ) e. B } =/= (/) )
8		sorpsscmpl 6948	. . . . . 6 |- ( [ C.] Or B -> [ C.] Or { c e. ~P A | ( A \ c ) e. B } )
9	8	ad2antll 765	. . . . 5 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> [ C.] Or { c e. ~P A | ( A \ c ) e. B } )
10		fin2i 9117	. . . . 5 |- ( ( ( A e. FinII /\ { c e. ~P A | ( A \ c ) e. B } C_ ~P A ) /\ ( { c e. ~P A | ( A \ c ) e. B } =/= (/) /\ [ C.] Or { c e. ~P A | ( A \ c ) e. B } ) ) -> U. { c e. ~P A | ( A \ c ) e. B } e. { c e. ~P A | ( A \ c ) e. B } )
11	2, 4, 7, 9, 10	syl22anc 1327	. . . 4 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> U. { c e. ~P A | ( A \ c ) e. B } e. { c e. ~P A | ( A \ c ) e. B } )
12		sorpssuni 6946	. . . . 5 |- ( [ C.] Or { c e. ~P A | ( A \ c ) e. B } -> ( E. m e. { c e. ~P A | ( A \ c ) e. B } A. n e. { c e. ~P A | ( A \ c ) e. B } -. m C. n <-> U. { c e. ~P A | ( A \ c ) e. B } e. { c e. ~P A | ( A \ c ) e. B } ) )
13	9, 12	syl 17	. . . 4 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> ( E. m e. { c e. ~P A | ( A \ c ) e. B } A. n e. { c e. ~P A | ( A \ c ) e. B } -. m C. n <-> U. { c e. ~P A | ( A \ c ) e. B } e. { c e. ~P A | ( A \ c ) e. B } ) )
14	11, 13	mpbird 247	. . 3 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> E. m e. { c e. ~P A | ( A \ c ) e. B } A. n e. { c e. ~P A | ( A \ c ) e. B } -. m C. n )
15		psseq2 3695	. . . 4 |- ( z = ( A \ m ) -> ( w C. z <-> w C. ( A \ m ) ) )
16		psseq2 3695	. . . 4 |- ( n = ( A \ w ) -> ( m C. n <-> m C. ( A \ w ) ) )
17		pssdifcom2 4055	. . . 4 |- ( ( m C_ A /\ w C_ A ) -> ( w C. ( A \ m ) <-> m C. ( A \ w ) ) )
18	15, 16, 17	fin23lem11 9139	. . 3 |- ( B C_ ~P A -> ( E. m e. { c e. ~P A | ( A \ c ) e. B } A. n e. { c e. ~P A | ( A \ c ) e. B } -. m C. n -> E. z e. B A. w e. B -. w C. z ) )
19	1, 14, 18	sylc 65	. 2 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> E. z e. B A. w e. B -. w C. z )
20		sorpssint 6947	. . 3 |- ( [ C.] Or B -> ( E. z e. B A. w e. B -. w C. z <-> |^| B e. B ) )
21	20	ad2antll 765	. 2 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> ( E. z e. B A. w e. B -. w C. z <-> |^| B e. B ) )
22	19, 21	mpbid 222	1 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> |^| B e. B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   Or wor 5034   [ C.] crpss 6936  Fin^IIcfin2 9101

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-pow 4843  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-rpss 6937  df-fin2 9108

This theorem is referenced by:  isfin2-2  9141  fin23lem40  9173  fin2so  33396