Theorem fin67 8775

Description: Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
fin67	|- ( A e. FinVI -> A e. FinVII )

Proof of Theorem fin67

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfin6 8680	. 2 |- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )
2		2onn 7289	. . . . . 6 |- 2o e. om
3		ssid 3523	. . . . . 6 |- 2o C_ 2o
4		ssnnfi 7739	. . . . . 6 |- ( ( 2o e. om /\ 2o C_ 2o ) -> 2o e. Fin )
5	2, 3, 4	mp2an 672	. . . . 5 |- 2o e. Fin
6		sdomdom 7543	. . . . 5 |- ( A ~< 2o -> A ~<_ 2o )
7		domfi 7741	. . . . 5 |- ( ( 2o e. Fin /\ A ~<_ 2o ) -> A e. Fin )
8	5, 6, 7	sylancr 663	. . . 4 |- ( A ~< 2o -> A e. Fin )
9		fin17 8774	. . . 4 |- ( A e. Fin -> A e. FinVII )
10	8, 9	syl 16	. . 3 |- ( A ~< 2o -> A e. FinVII )
11		sdomnen 7544	. . . . 5 |- ( A ~< ( A X. A ) -> -. A ~~ ( A X. A ) )
12		eldifi 3626	. . . . . . . . 9 |- ( b e. ( On \ om ) -> b e. On )
13		ensym 7564	. . . . . . . . 9 |- ( A ~~ b -> b ~~ A )
14		isnumi 8327	. . . . . . . . 9 |- ( ( b e. On /\ b ~~ A ) -> A e. dom card )
15	12, 13, 14	syl2an 477	. . . . . . . 8 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> A e. dom card )
16		vex 3116	. . . . . . . . . . 11 |- b e. _V
17		eldif 3486	. . . . . . . . . . . 12 |- ( b e. ( On \ om ) <-> ( b e. On /\ -. b e. om ) )
18		ordom 6693	. . . . . . . . . . . . . 14 |- Ord om
19		eloni 4888	. . . . . . . . . . . . . 14 |- ( b e. On -> Ord b )
20		ordtri1 4911	. . . . . . . . . . . . . 14 |- ( ( Ord om /\ Ord b ) -> ( om C_ b <-> -. b e. om ) )
21	18, 19, 20	sylancr 663	. . . . . . . . . . . . 13 |- ( b e. On -> ( om C_ b <-> -. b e. om ) )
22	21	biimpar 485	. . . . . . . . . . . 12 |- ( ( b e. On /\ -. b e. om ) -> om C_ b )
23	17, 22	sylbi 195	. . . . . . . . . . 11 |- ( b e. ( On \ om ) -> om C_ b )
24		ssdomg 7561	. . . . . . . . . . 11 |- ( b e. _V -> ( om C_ b -> om ~<_ b ) )
25	16, 23, 24	mpsyl 63	. . . . . . . . . 10 |- ( b e. ( On \ om ) -> om ~<_ b )
26		domen2 7660	. . . . . . . . . 10 |- ( A ~~ b -> ( om ~<_ A <-> om ~<_ b ) )
27	25, 26	syl5ibr 221	. . . . . . . . 9 |- ( A ~~ b -> ( b e. ( On \ om ) -> om ~<_ A ) )
28	27	impcom 430	. . . . . . . 8 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> om ~<_ A )
29		infxpidm2 8394	. . . . . . . 8 |- ( ( A e. dom card /\ om ~<_ A ) -> ( A X. A ) ~~ A )
30	15, 28, 29	syl2anc 661	. . . . . . 7 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> ( A X. A ) ~~ A )
31		ensym 7564	. . . . . . 7 |- ( ( A X. A ) ~~ A -> A ~~ ( A X. A ) )
32	30, 31	syl 16	. . . . . 6 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> A ~~ ( A X. A ) )
33	32	rexlimiva 2951	. . . . 5 |- ( E. b e. ( On \ om ) A ~~ b -> A ~~ ( A X. A ) )
34	11, 33	nsyl 121	. . . 4 |- ( A ~< ( A X. A ) -> -. E. b e. ( On \ om ) A ~~ b )
35		relsdom 7523	. . . . . 6 |- Rel ~<
36	35	brrelexi 5040	. . . . 5 |- ( A ~< ( A X. A ) -> A e. _V )
37		isfin7 8681	. . . . 5 |- ( A e. _V -> ( A e. FinVII <-> -. E. b e. ( On \ om ) A ~~ b ) )
38	36, 37	syl 16	. . . 4 |- ( A ~< ( A X. A ) -> ( A e. FinVII <-> -. E. b e. ( On \ om ) A ~~ b ) )
39	34, 38	mpbird 232	. . 3 |- ( A ~< ( A X. A ) -> A e. FinVII )
40	10, 39	jaoi 379	. 2 |- ( ( A ~< 2o \/ A ~< ( A X. A ) ) -> A e. FinVII )
41	1, 40	sylbi 195	1 |- ( A e. FinVI -> A e. FinVII )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   e. wcel 1767   E.wrex 2815   _Vcvv 3113   \ cdif 3473   C_ wss 3476   class class class wbr 4447   Ord word 4877   Oncon0 4878   X. cxp 4997   dom cdm 4999   omcom 6684   2oc2o 7124   ~~ cen 7513   ~<_ cdom 7514   ~< csdm 7515   Fincfn 7516   cardccrd 8316  Fin^VIcfin6 8663  Fin^VIIcfin7 8664

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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058

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-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-card 8320  df-fin6 8670  df-fin7 8671

This theorem is referenced by:  fin2so  29645