Theorem inf0 4751

Description: Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class "ran (rec({<.v, u>. | u = suc v}, x) |` om)" exists, is a subset of its union, and contains a given set x (and thus is non-empty). Thus it provides an example demonstrating that a set y exists with the necessary properties demanded by ax-inf 4767.

Hypothesis

Ref	Expression
inf0.1	|- om e. V

Assertion

Ref	Expression
inf0	|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))

Distinct variable group:   x,y,z,w

Proof of Theorem inf0

Step	Hyp	Ref	Expression
1		visset 1859	. . . 4 |- x e. V
2		fr0g 4253	. . . 4 |- (x e. V -> ((rec({<.v, u>. | u = suc v}, x) |` om)` (/)) = x)
3	1, 2	ax-mp 7	. . 3 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` (/)) = x
4		frfnom 4252	. . . 4 |- (rec({<.v, u>. | u = suc v}, x) |` om) Fn om
5		peano1 3237	. . . 4 |- (/) e. om
6		fnfvelrn 3927	. . . 4 |- (((rec({<.v, u>. | u = suc v}, x) |` om) Fn om /\ (/) e. om) -> ((rec({<.v, u>. | u = suc v}, x) |` om)` (/)) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))
7	4, 5, 6	mp2an 701	. . 3 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` (/)) e. ran (rec({<.v, u>. | u = suc v}, x) |` om)
8	3, 7	eqeltrri 1588	. 2 |- x e. ran (rec({<.v, u>. | u = suc v}, x) |` om)
9		fvelrnb 3871	. . . . 5 |- ((rec({<.v, u>. | u = suc v}, x) |` om) Fn om -> (z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) <-> E.f e. om ((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z))
10	4, 9	ax-mp 7	. . . 4 |- (z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) <-> E.f e. om ((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z)
11		eleq1 1577	. . . . . . . 8 |- (((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> (((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) <-> z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f)))
12		fvex 3843	. . . . . . . . . . 11 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. V
13	12	sucex 3168	. . . . . . . . . 10 |- suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. V
14		ax-17 1007	. . . . . . . . . . 11 |- (z e. x -> A.v z e. x)
15		ax-17 1007	. . . . . . . . . . 11 |- (z e. f -> A.v z e. f)
16		hbopab1 2890	. . . . . . . . . . . . . . 15 |- (z e. {<.v, u>. | u = suc v} -> A.v z e. {<.v, u>. | u = suc v})
17	16, 14	hbrdg 4237	. . . . . . . . . . . . . 14 |- (z e. rec({<.v, u>. | u = suc v}, x) -> A.v z e. rec({<.v, u>. | u = suc v}, x))
18		ax-17 1007	. . . . . . . . . . . . . 14 |- (z e. om -> A.v z e. om)
19	17, 18	hbres 3457	. . . . . . . . . . . . 13 |- (z e. (rec({<.v, u>. | u = suc v}, x) |` om) -> A.v z e. (rec({<.v, u>. | u = suc v}, x) |` om))
20	19, 15	hbfv 3840	. . . . . . . . . . . 12 |- (z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` f) -> A.v z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` f))
21	20	hbsuc 3044	. . . . . . . . . . 11 |- (z e. suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f) -> A.v z e. suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f))
22		eqid 1518	. . . . . . . . . . 11 |- (rec({<.v, u>. | u = suc v}, x) |` om) = (rec({<.v, u>. | u = suc v}, x) |` om)
23		suceq 3038	. . . . . . . . . . 11 |- (v = ((rec({<.v, u>. | u = suc v}, x) |` om)` f) -> suc v = suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f))
24	14, 15, 21, 22, 23	frsucopab 4255	. . . . . . . . . 10 |- ((f e. om /\ suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. V) -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) = suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f))
25	13, 24	mpan2 700	. . . . . . . . 9 |- (f e. om -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) = suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f))
26	12	sucid 3051	. . . . . . . . 9 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f)
27	25, 26	syl5eleqr 1598	. . . . . . . 8 |- (f e. om -> ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f))
28	11, 27	syl5bi 206	. . . . . . 7 |- (((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> (f e. om -> z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f)))
29		peano2b 3234	. . . . . . . . 9 |- (f e. om <-> suc f e. om)
30		fnfvelrn 3927	. . . . . . . . . 10 |- (((rec({<.v, u>. | u = suc v}, x) |` om) Fn om /\ suc f e. om) -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))
31	4, 30	mpan 699	. . . . . . . . 9 |- (suc f e. om -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))
32	29, 31	sylbi 197	. . . . . . . 8 |- (f e. om -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))
33	32	a1i 8	. . . . . . 7 |- (((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> (f e. om -> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
34	28, 33	jcad 603	. . . . . 6 |- (((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> (f e. om -> (z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) /\ ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))
35		fvex 3843	. . . . . . 7 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. V
36		eleq2 1578	. . . . . . . 8 |- (w = ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) -> (z e. w <-> z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f)))
37		eleq1 1577	. . . . . . . 8 |- (w = ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) -> (w e. ran (rec({<.v, u>. | u = suc v}, x) |` om) <-> ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
38	36, 37	anbi12d 631	. . . . . . 7 |- (w = ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) -> ((z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)) <-> (z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) /\ ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))
39	35, 38	cla4ev 1915	. . . . . 6 |- ((z e. ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) /\ ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. ran (rec({<.v, u>. | u = suc v}, x) |` om)) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
40	34, 39	syl6com 53	. . . . 5 |- (f e. om -> (((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))
41	40	r19.23aiv 1789	. . . 4 |- (E.f e. om ((rec({<.v, u>. | u = suc v}, x) |` om)` f) = z -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
42	10, 41	sylbi 197	. . 3 |- (z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
43	42	ax-gen 999	. 2 |- A.z(z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
44		fndm 3693	. . . . . 6 |- ((rec({<.v, u>. | u = suc v}, x) |` om) Fn om -> dom (rec({<.v, u>. | u = suc v}, x) |` om) = om)
45	4, 44	ax-mp 7	. . . . 5 |- dom (rec({<.v, u>. | u = suc v}, x) |` om) = om
46		inf0.1	. . . . 5 |- om e. V
47	45, 46	eqeltri 1587	. . . 4 |- dom (rec({<.v, u>. | u = suc v}, x) |` om) e. V
48		fnfun 3691	. . . . 5 |- ((rec({<.v, u>. | u = suc v}, x) |` om) Fn om -> Fun (rec({<.v, u>. | u = suc v}, x) |` om))
49	4, 48	ax-mp 7	. . . 4 |- Fun (rec({<.v, u>. | u = suc v}, x) |` om)
50		funrnex 3720	. . . 4 |- (dom (rec({<.v, u>. | u = suc v}, x) |` om) e. V -> (Fun (rec({<.v, u>. | u = suc v}, x) |` om) -> ran (rec({<.v, u>. | u = suc v}, x) |` om) e. V))
51	47, 49, 50	mp2 43	. . 3 |- ran (rec({<.v, u>. | u = suc v}, x) |` om) e. V
52		eleq2 1578	. . . 4 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> (x e. y <-> x e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
53		eleq2 1578	. . . . . 6 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> (z e. y <-> z e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
54		eleq2 1578	. . . . . . . 8 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> (w e. y <-> w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))
55	54	anbi2d 619	. . . . . . 7 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> ((z e. w /\ w e. y) <-> (z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))
56	55	exbidv 1317	. . . . . 6 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> (E.w(z e. w /\ w e. y) <-> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))
57	53, 56	imbi12d 629	. . . . 5 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> ((z e. y -> E.w(z e. w /\ w e. y)) <-> (z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))))
58	57	albidv 1316	. . . 4 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> (A.z(z e. y -> E.w(z e. w /\ w e. y)) <-> A.z(z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))))
59	52, 58	anbi12d 631	. . 3 |- (y = ran (rec({<.v, u>. | u = suc v}, x) |` om) -> ((x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y))) <-> (x e. ran (rec({<.v, u>. | u = suc v}, x) |` om) /\ A.z(z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om))))))
60	51, 59	cla4ev 1915	. 2 |- ((x e. ran (rec({<.v, u>. | u = suc v}, x) |` om) /\ A.z(z e. ran (rec({<.v, u>. | u = suc v}, x) |` om) -> E.w(z e. w /\ w e. ran (rec({<.v, u>. | u = suc v}, x) |` om)))) -> E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y))))
61	8, 43, 60	mp2an 701	1 |- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016  E.wrex 1692  Vcvv 1857  (/)c0 2332  {copab 2740  suc csuc 2977  omcom 3218  dom cdm 3251  ran crn 3252   |` cres 3253  Fun wfun 3257   Fn wfn 3258  ` cfv 3263  reccrdg 4232

This theorem is referenced by:  axinf 4773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279  df-rdg 4233

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