Theorem dcomex 9269

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	|- om e. _V

Proof of Theorem dcomex

Dummy variables t s x f n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 7575	. . . . . . 7 |- 1o =/= (/)
2		df-br 4654	. . . . . . . 8 |- ( ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) <-> <. ( f ` n ) , ( f ` suc n ) >. e. { <. 1o , 1o >. } )
3		elsni 4194	. . . . . . . . 9 |- ( <. ( f ` n ) , ( f ` suc n ) >. e. { <. 1o , 1o >. } -> <. ( f ` n ) , ( f ` suc n ) >. = <. 1o , 1o >. )
4		fvex 6201	. . . . . . . . . 10 |- ( f ` n ) e. _V
5		fvex 6201	. . . . . . . . . 10 |- ( f ` suc n ) e. _V
6	4, 5	opth1 4944	. . . . . . . . 9 |- ( <. ( f ` n ) , ( f ` suc n ) >. = <. 1o , 1o >. -> ( f ` n ) = 1o )
7	3, 6	syl 17	. . . . . . . 8 |- ( <. ( f ` n ) , ( f ` suc n ) >. e. { <. 1o , 1o >. } -> ( f ` n ) = 1o )
8	2, 7	sylbi 207	. . . . . . 7 |- ( ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> ( f ` n ) = 1o )
9		tz6.12i 6214	. . . . . . 7 |- ( 1o =/= (/) -> ( ( f ` n ) = 1o -> n f 1o ) )
10	1, 8, 9	mpsyl 68	. . . . . 6 |- ( ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> n f 1o )
11		vex 3203	. . . . . . 7 |- n e. _V
12		1on 7567	. . . . . . . 8 |- 1o e. On
13	12	elexi 3213	. . . . . . 7 |- 1o e. _V
14	11, 13	breldm 5329	. . . . . 6 |- ( n f 1o -> n e. dom f )
15	10, 14	syl 17	. . . . 5 |- ( ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> n e. dom f )
16	15	ralimi 2952	. . . 4 |- ( A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> A. n e. om n e. dom f )
17		dfss3 3592	. . . 4 |- ( om C_ dom f <-> A. n e. om n e. dom f )
18	16, 17	sylibr 224	. . 3 |- ( A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> om C_ dom f )
19		vex 3203	. . . . 5 |- f e. _V
20	19	dmex 7099	. . . 4 |- dom f e. _V
21	20	ssex 4802	. . 3 |- ( om C_ dom f -> om e. _V )
22	18, 21	syl 17	. 2 |- ( A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) -> om e. _V )
23		snex 4908	. . 3 |- { <. 1o , 1o >. } e. _V
24	13, 13	fvsn 6446	. . . . . . . 8 |- ( { <. 1o , 1o >. } ` 1o ) = 1o
25	13, 13	funsn 5939	. . . . . . . . 9 |- Fun { <. 1o , 1o >. }
26	13	snid 4208	. . . . . . . . . 10 |- 1o e. { 1o }
27	13	dmsnop 5609	. . . . . . . . . 10 |- dom { <. 1o , 1o >. } = { 1o }
28	26, 27	eleqtrri 2700	. . . . . . . . 9 |- 1o e. dom { <. 1o , 1o >. }
29		funbrfvb 6238	. . . . . . . . 9 |- ( ( Fun { <. 1o , 1o >. } /\ 1o e. dom { <. 1o , 1o >. } ) -> ( ( { <. 1o , 1o >. } ` 1o ) = 1o <-> 1o { <. 1o , 1o >. } 1o ) )
30	25, 28, 29	mp2an 708	. . . . . . . 8 |- ( ( { <. 1o , 1o >. } ` 1o ) = 1o <-> 1o { <. 1o , 1o >. } 1o )
31	24, 30	mpbi 220	. . . . . . 7 |- 1o { <. 1o , 1o >. } 1o
32		breq12 4658	. . . . . . . 8 |- ( ( s = 1o /\ t = 1o ) -> ( s { <. 1o , 1o >. } t <-> 1o { <. 1o , 1o >. } 1o ) )
33	13, 13, 32	spc2ev 3301	. . . . . . 7 |- ( 1o { <. 1o , 1o >. } 1o -> E. s E. t s { <. 1o , 1o >. } t )
34	31, 33	ax-mp 5	. . . . . 6 |- E. s E. t s { <. 1o , 1o >. } t
35		breq 4655	. . . . . . 7 |- ( x = { <. 1o , 1o >. } -> ( s x t <-> s { <. 1o , 1o >. } t ) )
36	35	2exbidv 1852	. . . . . 6 |- ( x = { <. 1o , 1o >. } -> ( E. s E. t s x t <-> E. s E. t s { <. 1o , 1o >. } t ) )
37	34, 36	mpbiri 248	. . . . 5 |- ( x = { <. 1o , 1o >. } -> E. s E. t s x t )
38		ssid 3624	. . . . . . 7 |- { 1o } C_ { 1o }
39	13	rnsnop 5616	. . . . . . 7 |- ran { <. 1o , 1o >. } = { 1o }
40	38, 39, 27	3sstr4i 3644	. . . . . 6 |- ran { <. 1o , 1o >. } C_ dom { <. 1o , 1o >. }
41		rneq 5351	. . . . . . 7 |- ( x = { <. 1o , 1o >. } -> ran x = ran { <. 1o , 1o >. } )
42		dmeq 5324	. . . . . . 7 |- ( x = { <. 1o , 1o >. } -> dom x = dom { <. 1o , 1o >. } )
43	41, 42	sseq12d 3634	. . . . . 6 |- ( x = { <. 1o , 1o >. } -> ( ran x C_ dom x <-> ran { <. 1o , 1o >. } C_ dom { <. 1o , 1o >. } ) )
44	40, 43	mpbiri 248	. . . . 5 |- ( x = { <. 1o , 1o >. } -> ran x C_ dom x )
45		pm5.5 351	. . . . 5 |- ( ( E. s E. t s x t /\ ran x C_ dom x ) -> ( ( ( E. s E. t s x t /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) ) <-> E. f A. n e. om ( f ` n ) x ( f ` suc n ) ) )
46	37, 44, 45	syl2anc 693	. . . 4 |- ( x = { <. 1o , 1o >. } -> ( ( ( E. s E. t s x t /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) ) <-> E. f A. n e. om ( f ` n ) x ( f ` suc n ) ) )
47		breq 4655	. . . . . 6 |- ( x = { <. 1o , 1o >. } -> ( ( f ` n ) x ( f ` suc n ) <-> ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) ) )
48	47	ralbidv 2986	. . . . 5 |- ( x = { <. 1o , 1o >. } -> ( A. n e. om ( f ` n ) x ( f ` suc n ) <-> A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) ) )
49	48	exbidv 1850	. . . 4 |- ( x = { <. 1o , 1o >. } -> ( E. f A. n e. om ( f ` n ) x ( f ` suc n ) <-> E. f A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) ) )
50	46, 49	bitrd 268	. . 3 |- ( x = { <. 1o , 1o >. } -> ( ( ( E. s E. t s x t /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) ) <-> E. f A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n ) ) )
51		ax-dc 9268	. . 3 |- ( ( E. s E. t s x t /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) )
52	23, 50, 51	vtocl 3259	. 2 |- E. f A. n e. om ( f ` n ) { <. 1o , 1o >. } ( f ` suc n )
53	22, 52	exlimiiv 1859	1 |- om e. _V

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   ran crn 5115   Oncon0 5723   suc csuc 5725   Fun wfun 5882   ` cfv 5888   omcom 7065   1oc1o 7553

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-pr 4906  ax-un 6949  ax-dc 9268

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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-1o 7560

This theorem is referenced by:  axdc2lem  9270  axdc3lem  9272  axdc4lem  9277  axcclem  9279