Theorem inf0 5712

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) " exi sts, 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 5728.

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 2295	. . . 4 |- x e. _V
2		fr0g 5160	. . . 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 5159	. . . 4 |- (rec({<.v, u>. | u = suc v}, x) |` om) Fn om
5		peano1 3971	. . . 4 |- (/) e. om
6		fnfvelrn 4786	. . . 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 761	. . 3 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` (/)) e. ran (rec({<.v, u>. | u = suc v}, x) |` om)
8	3, 7	eqeltrri 1968	. 2 |- x e. ran (rec({<.v, u>. | u = suc v}, x) |` om)
9		fvelrnb 4719	. . . . 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 1957	. . . . . . . 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 4689	. . . . . . . . . . 11 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. _V
13	12	sucex 3892	. . . . . . . . . 10 |- suc ((rec({<.v, u>. | u = suc v}, x) |` om)` f) e. _V
14		ax-17 1317	. . . . . . . . . . 11 |- (z e. x -> A.v z e. x)
15		ax-17 1317	. . . . . . . . . . 11 |- (z e. f -> A.v z e. f)
16		hbopab1 3562	. . . . . . . . . . . . . . 15 |- (z e. {<.v, u>. | u = suc v} -> A.v z e. {<.v, u>. | u = suc v})
17	16, 14	hbrdg 5144	. . . . . . . . . . . . . 14 |- (z e. rec({<.v, u>. | u = suc v}, x) -> A.v z e. rec({<.v, u>. | u = suc v}, x))
18		ax-17 1317	. . . . . . . . . . . . . 14 |- (z e. om -> A.v z e. om)
19	17, 18	hbres 4220	. . . . . . . . . . . . 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 4686	. . . . . . . . . . . 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 3736	. . . . . . . . . . 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 1884	. . . . . . . . . . 11 |- (rec({<.v, u>. | u = suc v}, x) |` om) = (rec({<.v, u>. | u = suc v}, x) |` om)
23		suceq 3729	. . . . . . . . . . 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 5162	. . . . . . . . . 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 760	. . . . . . . . 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 3744	. . . . . . . . 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 1978	. . . . . . . 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 225	. . . . . . 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 3968	. . . . . . . . 9 |- (f e. om <-> suc f e. om)
30		fnfvelrn 4786	. . . . . . . . . 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 759	. . . . . . . . 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 216	. . . . . . . 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 661	. . . . . 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 4689	. . . . . . 7 |- ((rec({<.v, u>. | u = suc v}, x) |` om)` suc f) e. _V
36		eleq2 1958	. . . . . . . 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 1957	. . . . . . . 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 690	. . . . . . 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 2371	. . . . . 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 64	. . . . 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 2211	. . . 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 216	. . 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 1305	. 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 4512	. . . . . 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 1967	. . . 4 |- dom (rec({<.v, u>. | u = suc v}, x) |` om) e. _V
48		fnfun 4510	. . . . 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 4544	. . . 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 54	. . 3 |- ran (rec({<.v, u>. | u = suc v}, x) |` om) e. _V
52		eleq2 1958	. . . 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 1958	. . . . . 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 1958	. . . . . . . 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 678	. . . . . . 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 1657	. . . . . 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 688	. . . . 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 1656	. . . 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 690	. . 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 2371	. 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 761	1 |- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292  (/)c0 2875  {copab 3395  suc csuc 3659  omcom 3949  dom cdm 3986  ran crn 3987   |` cres 3988  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  reccrdg 5139

This theorem is referenced by:  axinf 5734

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

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