Theorem fin67 9217

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 9122	. 2 |- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )
2		2onn 7720	. . . . . 6 |- 2o e. om
3		ssid 3624	. . . . . 6 |- 2o C_ 2o
4		ssnnfi 8179	. . . . . 6 |- ( ( 2o e. om /\ 2o C_ 2o ) -> 2o e. Fin )
5	2, 3, 4	mp2an 708	. . . . 5 |- 2o e. Fin
6		sdomdom 7983	. . . . 5 |- ( A ~< 2o -> A ~<_ 2o )
7		domfi 8181	. . . . 5 |- ( ( 2o e. Fin /\ A ~<_ 2o ) -> A e. Fin )
8	5, 6, 7	sylancr 695	. . . 4 |- ( A ~< 2o -> A e. Fin )
9		fin17 9216	. . . 4 |- ( A e. Fin -> A e. FinVII )
10	8, 9	syl 17	. . 3 |- ( A ~< 2o -> A e. FinVII )
11		sdomnen 7984	. . . . 5 |- ( A ~< ( A X. A ) -> -. A ~~ ( A X. A ) )
12		eldifi 3732	. . . . . . . . 9 |- ( b e. ( On \ om ) -> b e. On )
13		ensym 8005	. . . . . . . . 9 |- ( A ~~ b -> b ~~ A )
14		isnumi 8772	. . . . . . . . 9 |- ( ( b e. On /\ b ~~ A ) -> A e. dom card )
15	12, 13, 14	syl2an 494	. . . . . . . 8 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> A e. dom card )
16		vex 3203	. . . . . . . . . . 11 |- b e. _V
17		eldif 3584	. . . . . . . . . . . 12 |- ( b e. ( On \ om ) <-> ( b e. On /\ -. b e. om ) )
18		ordom 7074	. . . . . . . . . . . . . 14 |- Ord om
19		eloni 5733	. . . . . . . . . . . . . 14 |- ( b e. On -> Ord b )
20		ordtri1 5756	. . . . . . . . . . . . . 14 |- ( ( Ord om /\ Ord b ) -> ( om C_ b <-> -. b e. om ) )
21	18, 19, 20	sylancr 695	. . . . . . . . . . . . 13 |- ( b e. On -> ( om C_ b <-> -. b e. om ) )
22	21	biimpar 502	. . . . . . . . . . . 12 |- ( ( b e. On /\ -. b e. om ) -> om C_ b )
23	17, 22	sylbi 207	. . . . . . . . . . 11 |- ( b e. ( On \ om ) -> om C_ b )
24		ssdomg 8001	. . . . . . . . . . 11 |- ( b e. _V -> ( om C_ b -> om ~<_ b ) )
25	16, 23, 24	mpsyl 68	. . . . . . . . . 10 |- ( b e. ( On \ om ) -> om ~<_ b )
26		domen2 8103	. . . . . . . . . 10 |- ( A ~~ b -> ( om ~<_ A <-> om ~<_ b ) )
27	25, 26	syl5ibr 236	. . . . . . . . 9 |- ( A ~~ b -> ( b e. ( On \ om ) -> om ~<_ A ) )
28	27	impcom 446	. . . . . . . 8 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> om ~<_ A )
29		infxpidm2 8840	. . . . . . . 8 |- ( ( A e. dom card /\ om ~<_ A ) -> ( A X. A ) ~~ A )
30	15, 28, 29	syl2anc 693	. . . . . . 7 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> ( A X. A ) ~~ A )
31		ensym 8005	. . . . . . 7 |- ( ( A X. A ) ~~ A -> A ~~ ( A X. A ) )
32	30, 31	syl 17	. . . . . 6 |- ( ( b e. ( On \ om ) /\ A ~~ b ) -> A ~~ ( A X. A ) )
33	32	rexlimiva 3028	. . . . 5 |- ( E. b e. ( On \ om ) A ~~ b -> A ~~ ( A X. A ) )
34	11, 33	nsyl 135	. . . 4 |- ( A ~< ( A X. A ) -> -. E. b e. ( On \ om ) A ~~ b )
35		relsdom 7962	. . . . . 6 |- Rel ~<
36	35	brrelexi 5158	. . . . 5 |- ( A ~< ( A X. A ) -> A e. _V )
37		isfin7 9123	. . . . 5 |- ( A e. _V -> ( A e. FinVII <-> -. E. b e. ( On \ om ) A ~~ b ) )
38	36, 37	syl 17	. . . 4 |- ( A ~< ( A X. A ) -> ( A e. FinVII <-> -. E. b e. ( On \ om ) A ~~ b ) )
39	34, 38	mpbird 247	. . 3 |- ( A ~< ( A X. A ) -> A e. FinVII )
40	10, 39	jaoi 394	. 2 |- ( ( A ~< 2o \/ A ~< ( A X. A ) ) -> A e. FinVII )
41	1, 40	sylbi 207	1 |- ( A e. FinVI -> A e. FinVII )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   e. wcel 1990   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Ord word 5722   Oncon0 5723   omcom 7065   2oc2o 7554   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761  Fin^VIcfin6 9105  Fin^VIIcfin7 9106

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-fin6 9112  df-fin7 9113

This theorem is referenced by:  fin2so  33396