Theorem fin45 8557

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 8480	. . 3 |- ( A e. FinIV <-> A ~< ( A +c 1o ) )
2		simpl 454	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A =/= (/) )
3		relen 7311	. . . . . . . . . . . 12 |- Rel ~~
4	3	brrelexi 4875	. . . . . . . . . . 11 |- ( A ~~ ( A +c A ) -> A e. _V )
5	4	adantl 463	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A e. _V )
6		0sdomg 7436	. . . . . . . . . 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 7506	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
10	8, 9	sylib 196	. . . . . . 7 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> 1o ~<_ A )
11		cdadom2 8352	. . . . . . 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 7450	. . . . . . 7 |- ( A ~~ ( A +c A ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
14	13	adantl 463	. . . . . 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 7433	. . . . 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 8556	. 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 1761   =/= wne 2604   _Vcvv 2970   (/)c0 3634   class class class wbr 4289  (class class class)co 6090   1oc1o 6909   ~~ cen 7303   ~<_ cdom 7304   ~< csdm 7305   +c ccda 8332  Fin^IVcfin4 8445  Fin^Vcfin5 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-cda 8333  df-fin4 8452  df-fin5 8454

This theorem is referenced by:  fin2so  28341