Theorem fin45 8784

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 8707	. . 3 |- ( A e. FinIV <-> A ~< ( A +c 1o ) )
2		simpl 457	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A =/= (/) )
3		relen 7533	. . . . . . . . . . . 12 |- Rel ~~
4	3	brrelexi 5046	. . . . . . . . . . 11 |- ( A ~~ ( A +c A ) -> A e. _V )
5	4	adantl 466	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A e. _V )
6		0sdomg 7658	. . . . . . . . . 10 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
7	5, 6	syl 16	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( (/) ~< A <-> A =/= (/) ) )
8	2, 7	mpbird 232	. . . . . . . 8 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> (/) ~< A )
9		0sdom1dom 7729	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
10	8, 9	sylib 196	. . . . . . 7 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> 1o ~<_ A )
11		cdadom2 8579	. . . . . . 7 |- ( 1o ~<_ A -> ( A +c 1o ) ~<_ ( A +c A ) )
12	10, 11	syl 16	. . . . . 6 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( A +c 1o ) ~<_ ( A +c A ) )
13		domen2 7672	. . . . . . 7 |- ( A ~~ ( A +c A ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
14	13	adantl 466	. . . . . 6 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
15	12, 14	mpbird 232	. . . . 5 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( A +c 1o ) ~<_ A )
16		domnsym 7655	. . . . 5 |- ( ( A +c 1o ) ~<_ A -> -. A ~< ( A +c 1o ) )
17	15, 16	syl 16	. . . 4 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> -. A ~< ( A +c 1o ) )
18	17	con2i 120	. . 3 |- ( A ~< ( A +c 1o ) -> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) )
19	1, 18	sylbi 195	. 2 |- ( A e. FinIV -> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) )
20		isfin5-2 8783	. 2 |- ( A e. FinIV -> ( A e. FinV <-> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) ) )
21	19, 20	mpbird 232	1 |- ( A e. FinIV -> A e. FinV )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   =/= wne 2662   _Vcvv 3118   (/)c0 3790   class class class wbr 4453  (class class class)co 6295   1oc1o 7135   ~~ cen 7525   ~<_ cdom 7526   ~< csdm 7527   +c ccda 8559  Fin^IVcfin4 8672  Fin^Vcfin5 8674

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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-cda 8560  df-fin4 8679  df-fin5 8681

This theorem is referenced by:  fin2so  29974