Theorem fin45 8576

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 8499	. . 3 |- ( A e. FinIV <-> A ~< ( A +c 1o ) )
2		simpl 457	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A =/= (/) )
3		relen 7330	. . . . . . . . . . . 12 |- Rel ~~
4	3	brrelexi 4894	. . . . . . . . . . 11 |- ( A ~~ ( A +c A ) -> A e. _V )
5	4	adantl 466	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A e. _V )
6		0sdomg 7455	. . . . . . . . . 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 7525	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
10	8, 9	sylib 196	. . . . . . 7 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> 1o ~<_ A )
11		cdadom2 8371	. . . . . . 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 7469	. . . . . . 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 7452	. . . . 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 8575	. 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 1756   =/= wne 2620   _Vcvv 2987   (/)c0 3652   class class class wbr 4307  (class class class)co 6106   1oc1o 6928   ~~ cen 7322   ~<_ cdom 7323   ~< csdm 7324   +c ccda 8351  Fin^IVcfin4 8464  Fin^Vcfin5 8466

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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387

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 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-cda 8352  df-fin4 8471  df-fin5 8473

This theorem is referenced by:  fin2so  28435