Theorem fin45 8840

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 8763	. . 3 |- ( A e. FinIV <-> A ~< ( A +c 1o ) )
2		simpl 464	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A =/= (/) )
3		relen 7592	. . . . . . . . . . . 12 |- Rel ~~
4	3	brrelexi 4880	. . . . . . . . . . 11 |- ( A ~~ ( A +c A ) -> A e. _V )
5	4	adantl 473	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> A e. _V )
6		0sdomg 7719	. . . . . . . . . 10 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
7	5, 6	syl 17	. . . . . . . . 9 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( (/) ~< A <-> A =/= (/) ) )
8	2, 7	mpbird 240	. . . . . . . 8 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> (/) ~< A )
9		0sdom1dom 7788	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
10	8, 9	sylib 201	. . . . . . 7 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> 1o ~<_ A )
11		cdadom2 8635	. . . . . . 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 7733	. . . . . . 7 |- ( A ~~ ( A +c A ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
14	13	adantl 473	. . . . . 6 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( ( A +c 1o ) ~<_ A <-> ( A +c 1o ) ~<_ ( A +c A ) ) )
15	12, 14	mpbird 240	. . . . 5 |- ( ( A =/= (/) /\ A ~~ ( A +c A ) ) -> ( A +c 1o ) ~<_ A )
16		domnsym 7716	. . . . 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 200	. 2 |- ( A e. FinIV -> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) )
20		isfin5-2 8839	. 2 |- ( A e. FinIV -> ( A e. FinV <-> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) ) )
21	19, 20	mpbird 240	1 |- ( A e. FinIV -> A e. FinV )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   e. wcel 1904   =/= wne 2641   _Vcvv 3031   (/)c0 3722   class class class wbr 4395  (class class class)co 6308   1oc1o 7193   ~~ cen 7584   ~<_ cdom 7585   ~< csdm 7586   +c ccda 8615  Fin^IVcfin4 8728  Fin^Vcfin5 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-cda 8616  df-fin4 8735  df-fin5 8737

This theorem is referenced by:  fin2so  31996