Theorem cdainf 9014

Description: A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdainf	|- ( om ~<_ A <-> om ~<_ ( A +c A ) )

Proof of Theorem cdainf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . 5 |- Rel ~<_
2	1	brrelex2i 5159	. . . 4 |- ( om ~<_ A -> A e. _V )
3		cdadom3 9010	. . . 4 |- ( ( A e. _V /\ A e. _V ) -> A ~<_ ( A +c A ) )
4	2, 2, 3	syl2anc 693	. . 3 |- ( om ~<_ A -> A ~<_ ( A +c A ) )
5		domtr 8009	. . 3 |- ( ( om ~<_ A /\ A ~<_ ( A +c A ) ) -> om ~<_ ( A +c A ) )
6	4, 5	mpdan 702	. 2 |- ( om ~<_ A -> om ~<_ ( A +c A ) )
7		infn0 8222	. . . 4 |- ( om ~<_ ( A +c A ) -> ( A +c A ) =/= (/) )
8		cdafn 8991	. . . . . . . 8 |- +c Fn ( _V X. _V )
9		fndm 5990	. . . . . . . 8 |- ( +c Fn ( _V X. _V ) -> dom +c = ( _V X. _V ) )
10	8, 9	ax-mp 5	. . . . . . 7 |- dom +c = ( _V X. _V )
11	10	ndmov 6818	. . . . . 6 |- ( -. ( A e. _V /\ A e. _V ) -> ( A +c A ) = (/) )
12	11	necon1ai 2821	. . . . 5 |- ( ( A +c A ) =/= (/) -> ( A e. _V /\ A e. _V ) )
13	12	simpld 475	. . . 4 |- ( ( A +c A ) =/= (/) -> A e. _V )
14	7, 13	syl 17	. . 3 |- ( om ~<_ ( A +c A ) -> A e. _V )
15		ovex 6678	. . . . 5 |- ( A +c A ) e. _V
16	15	domen 7968	. . . 4 |- ( om ~<_ ( A +c A ) <-> E. x ( om ~~ x /\ x C_ ( A +c A ) ) )
17		indi 3873	. . . . . . . . 9 |- ( x i^i ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) ) = ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) )
18		simprr 796	. . . . . . . . . . 11 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> x C_ ( A +c A ) )
19		simpl 473	. . . . . . . . . . . 12 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> A e. _V )
20		cdaval 8992	. . . . . . . . . . . 12 |- ( ( A e. _V /\ A e. _V ) -> ( A +c A ) = ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) )
21	19, 19, 20	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> ( A +c A ) = ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) )
22	18, 21	sseqtrd 3641	. . . . . . . . . 10 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> x C_ ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) )
23		df-ss 3588	. . . . . . . . . 10 |- ( x C_ ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) <-> ( x i^i ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) ) = x )
24	22, 23	sylib 208	. . . . . . . . 9 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> ( x i^i ( ( A X. { (/) } ) u. ( A X. { 1o } ) ) ) = x )
25	17, 24	syl5eqr 2670	. . . . . . . 8 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) ) = x )
26		ensym 8005	. . . . . . . . 9 |- ( om ~~ x -> x ~~ om )
27	26	ad2antrl 764	. . . . . . . 8 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> x ~~ om )
28	25, 27	eqbrtrd 4675	. . . . . . 7 |- ( ( A e. _V /\ ( om ~~ x /\ x C_ ( A +c A ) ) ) -> ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) ) ~~ om )
29	28	ex 450	. . . . . 6 |- ( A e. _V -> ( ( om ~~ x /\ x C_ ( A +c A ) ) -> ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) ) ~~ om ) )
30		cdainflem 9013	. . . . . . 7 |- ( ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) ) ~~ om -> ( ( x i^i ( A X. { (/) } ) ) ~~ om \/ ( x i^i ( A X. { 1o } ) ) ~~ om ) )
31		snex 4908	. . . . . . . . . . . 12 |- { (/) } e. _V
32		xpexg 6960	. . . . . . . . . . . 12 |- ( ( A e. _V /\ { (/) } e. _V ) -> ( A X. { (/) } ) e. _V )
33	31, 32	mpan2 707	. . . . . . . . . . 11 |- ( A e. _V -> ( A X. { (/) } ) e. _V )
34		inss2 3834	. . . . . . . . . . 11 |- ( x i^i ( A X. { (/) } ) ) C_ ( A X. { (/) } )
35		ssdomg 8001	. . . . . . . . . . 11 |- ( ( A X. { (/) } ) e. _V -> ( ( x i^i ( A X. { (/) } ) ) C_ ( A X. { (/) } ) -> ( x i^i ( A X. { (/) } ) ) ~<_ ( A X. { (/) } ) ) )
36	33, 34, 35	mpisyl 21	. . . . . . . . . 10 |- ( A e. _V -> ( x i^i ( A X. { (/) } ) ) ~<_ ( A X. { (/) } ) )
37		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
38		xpsneng 8045	. . . . . . . . . . 11 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
39	37, 38	mpan2 707	. . . . . . . . . 10 |- ( A e. _V -> ( A X. { (/) } ) ~~ A )
40		domentr 8015	. . . . . . . . . 10 |- ( ( ( x i^i ( A X. { (/) } ) ) ~<_ ( A X. { (/) } ) /\ ( A X. { (/) } ) ~~ A ) -> ( x i^i ( A X. { (/) } ) ) ~<_ A )
41	36, 39, 40	syl2anc 693	. . . . . . . . 9 |- ( A e. _V -> ( x i^i ( A X. { (/) } ) ) ~<_ A )
42		domen1 8102	. . . . . . . . 9 |- ( ( x i^i ( A X. { (/) } ) ) ~~ om -> ( ( x i^i ( A X. { (/) } ) ) ~<_ A <-> om ~<_ A ) )
43	41, 42	syl5ibcom 235	. . . . . . . 8 |- ( A e. _V -> ( ( x i^i ( A X. { (/) } ) ) ~~ om -> om ~<_ A ) )
44		snex 4908	. . . . . . . . . . . 12 |- { 1o } e. _V
45		xpexg 6960	. . . . . . . . . . . 12 |- ( ( A e. _V /\ { 1o } e. _V ) -> ( A X. { 1o } ) e. _V )
46	44, 45	mpan2 707	. . . . . . . . . . 11 |- ( A e. _V -> ( A X. { 1o } ) e. _V )
47		inss2 3834	. . . . . . . . . . 11 |- ( x i^i ( A X. { 1o } ) ) C_ ( A X. { 1o } )
48		ssdomg 8001	. . . . . . . . . . 11 |- ( ( A X. { 1o } ) e. _V -> ( ( x i^i ( A X. { 1o } ) ) C_ ( A X. { 1o } ) -> ( x i^i ( A X. { 1o } ) ) ~<_ ( A X. { 1o } ) ) )
49	46, 47, 48	mpisyl 21	. . . . . . . . . 10 |- ( A e. _V -> ( x i^i ( A X. { 1o } ) ) ~<_ ( A X. { 1o } ) )
50		1on 7567	. . . . . . . . . . 11 |- 1o e. On
51		xpsneng 8045	. . . . . . . . . . 11 |- ( ( A e. _V /\ 1o e. On ) -> ( A X. { 1o } ) ~~ A )
52	50, 51	mpan2 707	. . . . . . . . . 10 |- ( A e. _V -> ( A X. { 1o } ) ~~ A )
53		domentr 8015	. . . . . . . . . 10 |- ( ( ( x i^i ( A X. { 1o } ) ) ~<_ ( A X. { 1o } ) /\ ( A X. { 1o } ) ~~ A ) -> ( x i^i ( A X. { 1o } ) ) ~<_ A )
54	49, 52, 53	syl2anc 693	. . . . . . . . 9 |- ( A e. _V -> ( x i^i ( A X. { 1o } ) ) ~<_ A )
55		domen1 8102	. . . . . . . . 9 |- ( ( x i^i ( A X. { 1o } ) ) ~~ om -> ( ( x i^i ( A X. { 1o } ) ) ~<_ A <-> om ~<_ A ) )
56	54, 55	syl5ibcom 235	. . . . . . . 8 |- ( A e. _V -> ( ( x i^i ( A X. { 1o } ) ) ~~ om -> om ~<_ A ) )
57	43, 56	jaod 395	. . . . . . 7 |- ( A e. _V -> ( ( ( x i^i ( A X. { (/) } ) ) ~~ om \/ ( x i^i ( A X. { 1o } ) ) ~~ om ) -> om ~<_ A ) )
58	30, 57	syl5 34	. . . . . 6 |- ( A e. _V -> ( ( ( x i^i ( A X. { (/) } ) ) u. ( x i^i ( A X. { 1o } ) ) ) ~~ om -> om ~<_ A ) )
59	29, 58	syld 47	. . . . 5 |- ( A e. _V -> ( ( om ~~ x /\ x C_ ( A +c A ) ) -> om ~<_ A ) )
60	59	exlimdv 1861	. . . 4 |- ( A e. _V -> ( E. x ( om ~~ x /\ x C_ ( A +c A ) ) -> om ~<_ A ) )
61	16, 60	syl5bi 232	. . 3 |- ( A e. _V -> ( om ~<_ ( A +c A ) -> om ~<_ A ) )
62	14, 61	mpcom 38	. 2 |- ( om ~<_ ( A +c A ) -> om ~<_ A )
63	6, 62	impbii 199	1 |- ( om ~<_ A <-> om ~<_ ( A +c A ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   Fn wfn 5883  (class class class)co 6650   omcom 7065   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   +c ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-cda 8990

This theorem is referenced by:  infdif  9031