Theorem fin67 8564

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 8469	. 2 |- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )
2		2onn 7079	. . . . . 6 |- 2o e. om
3		ssid 3375	. . . . . 6 |- 2o C_ 2o
4		ssnnfi 7532	. . . . . 6 |- ( ( 2o e. om /\ 2o C_ 2o ) -> 2o e. Fin )
5	2, 3, 4	mp2an 672	. . . . 5 |- 2o e. Fin
6		sdomdom 7337	. . . . 5 |- ( A ~< 2o -> A ~<_ 2o )
7		domfi 7534	. . . . 5 |- ( ( 2o e. Fin /\ A ~<_ 2o ) -> A e. Fin )
8	5, 6, 7	sylancr 663	. . . 4 |- ( A ~< 2o -> A e. Fin )
9		fin17 8563	. . . 4 |- ( A e. Fin -> A e. FinVII )
10	8, 9	syl 16	. . 3 |- ( A ~< 2o -> A e. FinVII )
11		sdomnen 7338	. . . . 5 |- ( A ~< ( A X. A ) -> -. A ~~ ( A X. A ) )
12		eldifi 3478	. . . . . . . . 9 |- ( b e. ( On \ om ) -> b e. On )
13		ensym 7358	. . . . . . . . 9 |- ( A ~~ b -> b ~~ A )
14		isnumi 8116	. . . . . . . . 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 2975	. . . . . . . . . . 11 |- b e. _V
17		eldif 3338	. . . . . . . . . . . 12 |- ( b e. ( On \ om ) <-> ( b e. On /\ -. b e. om ) )
18		ordom 6485	. . . . . . . . . . . . . 14 |- Ord om
19		eloni 4729	. . . . . . . . . . . . . 14 |- ( b e. On -> Ord b )
20		ordtri1 4752	. . . . . . . . . . . . . 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 7355	. . . . . . . . . . 11 |- ( b e. _V -> ( om C_ b -> om ~<_ b ) )
25	16, 23, 24	mpsyl 63	. . . . . . . . . 10 |- ( b e. ( On \ om ) -> om ~<_ b )
26		domen2 7454	. . . . . . . . . 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 8183	. . . . . . . 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 7358	. . . . . . 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 2836	. . . . 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 7317	. . . . . 6 |- Rel ~<
36	35	brrelexi 4879	. . . . 5 |- ( A ~< ( A X. A ) -> A e. _V )
37		isfin7 8470	. . . . 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 1756   E.wrex 2716   _Vcvv 2972   \ cdif 3325   C_ wss 3328   class class class wbr 4292   Ord word 4718   Oncon0 4719   X. cxp 4838   dom cdm 4840   omcom 6476   2oc2o 6914   ~~ cen 7307   ~<_ cdom 7308   ~< csdm 7309   Fincfn 7310   cardccrd 8105  Fin^VIcfin6 8452  Fin^VIIcfin7 8453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-card 8109  df-fin6 8459  df-fin7 8460

This theorem is referenced by:  fin2so  28416