Theorem fin45 8822

Description: Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
fin45	|- ( A e. FinIV -> A e. FinV )

Proof of Theorem fin45

Step	Hyp	Ref	Expression
1		isfin4-3 8745	. . 3 |- ( A e. FinIV <-> A ~< ( A +c 1o ) )
2		simpl 459	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A =/= (/) )
3		relen 7574	. . . . . . . . . . . 12 |- Rel ~~
4	3	brrelexi 4875	. . . . . . . . . . 11 |- ( A ~~ ( A +c A ) -> A e. _V )
5	4	adantl 468	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A e. _V )
6		0sdomg 7701	. . . . . . . . . 10 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
7	5, 6	syl 17	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( (/) ~< A <-> A =/= (/) ) )
8	2, 7	mpbird 236	. . . . . . . 8 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> (/) ~< A )
9		0sdom1dom 7770	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
10	8, 9	sylib 200	. . . . . . 7 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> 1o ~<_ A )
11		cdadom2 8617	. . . . . . 7 |- ( 1o ~<_ A -> ( A +c 1o ) ~<_ ( A +c A ) )
12	10, 11	syl 17	. . . . . 6 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( A +c 1o ) ~<_ ( A +c A ) )
13		domen2 7715	. . . . . . 7 |- ( A ~~ ( A +c A ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
14	13	adantl 468	. . . . . 6 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
15	12, 14	mpbird 236	. . . . 5 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( A +c 1o ) ~<_ A )
16		domnsym 7698	. . . . 5 |- ( ( A +c 1o ) ~<_ A -> -. A ~< ( A +c 1o ) )
17	15, 16	syl 17	. . . 4 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> -. A ~< ( A +c 1o ) )
18	17	con2i 124	. . 3 |- ( A ~< ( A +c 1o ) -> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) )
19	1, 18	sylbi 199	. 2 |- ( A e. FinIV -> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) )
20		isfin5-2 8821	. 2 |- ( A e. FinIV -> ( A e. FinV <-> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) ) )
21	19, 20	mpbird 236	1 |- ( A e. FinIV -> A e. FinV )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   e. wcel 1887   =/= wne 2622   _Vcvv 3045   (/)c0 3731   class class class wbr 4402  (class class class)co 6290   1oc1o 7175   ~~ cen 7566   ~<_ cdom 7567   ~< csdm 7568   +c ccda 8597  Fin^IVcfin4 8710  Fin^Vcfin5 8712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-cda 8598  df-fin4 8717  df-fin5 8719

This theorem is referenced by:  fin2so  31932