Theorem fin56 9215

Description: Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
fin56	|- ( A e. FinV -> A e. FinVI )

Proof of Theorem fin56

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( A = (/) -> ( A = (/) \/ A ~~ 1o ) )
2		sdom2en01 9124	. . . . 5 |- ( A ~< 2o <-> ( A = (/) \/ A ~~ 1o ) )
3	1, 2	sylibr 224	. . . 4 |- ( A = (/) -> A ~< 2o )
4	3	orcd 407	. . 3 |- ( A = (/) -> ( A ~< 2o \/ A ~< ( A X. A ) ) )
5		onfin2 8152	. . . . . . . 8 |- om = ( On i^i Fin )
6		inss2 3834	. . . . . . . 8 |- ( On i^i Fin ) C_ Fin
7	5, 6	eqsstri 3635	. . . . . . 7 |- om C_ Fin
8		2onn 7720	. . . . . . 7 |- 2o e. om
9	7, 8	sselii 3600	. . . . . 6 |- 2o e. Fin
10		relsdom 7962	. . . . . . 7 |- Rel ~<
11	10	brrelexi 5158	. . . . . 6 |- ( A ~< ( A +c A ) -> A e. _V )
12		fidomtri 8819	. . . . . 6 |- ( ( 2o e. Fin /\ A e. _V ) -> ( 2o ~<_ A <-> -. A ~< 2o ) )
13	9, 11, 12	sylancr 695	. . . . 5 |- ( A ~< ( A +c A ) -> ( 2o ~<_ A <-> -. A ~< 2o ) )
14		xp2cda 9002	. . . . . . . . . 10 |- ( A e. _V -> ( A X. 2o ) = ( A +c A ) )
15	11, 14	syl 17	. . . . . . . . 9 |- ( A ~< ( A +c A ) -> ( A X. 2o ) = ( A +c A ) )
16	15	adantr 481	. . . . . . . 8 |- ( ( A ~< ( A +c A ) /\ 2o ~<_ A ) -> ( A X. 2o ) = ( A +c A ) )
17		xpdom2g 8056	. . . . . . . . 9 |- ( ( A e. _V /\ 2o ~<_ A ) -> ( A X. 2o ) ~<_ ( A X. A ) )
18	11, 17	sylan 488	. . . . . . . 8 |- ( ( A ~< ( A +c A ) /\ 2o ~<_ A ) -> ( A X. 2o ) ~<_ ( A X. A ) )
19	16, 18	eqbrtrrd 4677	. . . . . . 7 |- ( ( A ~< ( A +c A ) /\ 2o ~<_ A ) -> ( A +c A ) ~<_ ( A X. A ) )
20		sdomdomtr 8093	. . . . . . 7 |- ( ( A ~< ( A +c A ) /\ ( A +c A ) ~<_ ( A X. A ) ) -> A ~< ( A X. A ) )
21	19, 20	syldan 487	. . . . . 6 |- ( ( A ~< ( A +c A ) /\ 2o ~<_ A ) -> A ~< ( A X. A ) )
22	21	ex 450	. . . . 5 |- ( A ~< ( A +c A ) -> ( 2o ~<_ A -> A ~< ( A X. A ) ) )
23	13, 22	sylbird 250	. . . 4 |- ( A ~< ( A +c A ) -> ( -. A ~< 2o -> A ~< ( A X. A ) ) )
24	23	orrd 393	. . 3 |- ( A ~< ( A +c A ) -> ( A ~< 2o \/ A ~< ( A X. A ) ) )
25	4, 24	jaoi 394	. 2 |- ( ( A = (/) \/ A ~< ( A +c A ) ) -> ( A ~< 2o \/ A ~< ( A X. A ) ) )
26		isfin5 9121	. 2 |- ( A e. FinV <-> ( A = (/) \/ A ~< ( A +c A ) ) )
27		isfin6 9122	. 2 |- ( A e. FinVI <-> ( A ~< 2o \/ A ~< ( A X. A ) ) )
28	25, 26, 27	3imtr4i 281	1 |- ( A e. FinV -> A e. FinVI )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   class class class wbr 4653   X. cxp 5112   Oncon0 5723  (class class class)co 6650   omcom 7065   1oc1o 7553   2oc2o 7554   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   +c ccda 8989  Fin^Vcfin5 9104  Fin^VIcfin6 9105

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-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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-fin5 9111  df-fin6 9112

This theorem is referenced by:  fin2so  33396