Theorem peano5 3238

Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction hypothesis, is derived from this theorem as theorem findes 3245.

Assertion

Ref	Expression
peano5	|- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)

Distinct variable group:   x,A

Proof of Theorem peano5

Step	Hyp	Ref	Expression
1		eldifn 2213	. . . . . 6 |- (y e. (om \ A) -> -. y e. A)
2	1	adantl 388	. . . . 5 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> -. y e. A)
3		nnsuc 3232	. . . . . . . . . 10 |- ((y e. om /\ y =/= (/)) -> E.x e. om y = suc x)
4		eldifi 2212	. . . . . . . . . . 11 |- (y e. (om \ A) -> y e. om)
5	4	adantl 388	. . . . . . . . . 10 |- (((/) e. A /\ y e. (om \ A)) -> y e. om)
6		eleq1 1575	. . . . . . . . . . . . . 14 |- (y = (/) -> (y e. (om \ A) <-> (/) e. (om \ A)))
7	6	biimpcd 153	. . . . . . . . . . . . 13 |- (y e. (om \ A) -> (y = (/) -> (/) e. (om \ A)))
8	7	necon3bd 1644	. . . . . . . . . . . 12 |- (y e. (om \ A) -> (-. (/) e. (om \ A) -> y =/= (/)))
9		elndif 2214	. . . . . . . . . . . 12 |- ((/) e. A -> -. (/) e. (om \ A))
10	8, 9	syl5com 52	. . . . . . . . . . 11 |- ((/) e. A -> (y e. (om \ A) -> y =/= (/)))
11	10	imp 348	. . . . . . . . . 10 |- (((/) e. A /\ y e. (om \ A)) -> y =/= (/))
12	3, 5, 11	sylanc 473	. . . . . . . . 9 |- (((/) e. A /\ y e. (om \ A)) -> E.x e. om y = suc x)
13	12	adantlr 393	. . . . . . . 8 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> E.x e. om y = suc x)
14	13	adantr 389	. . . . . . 7 |- (((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) /\ ((om \ A) i^i y) = (/)) -> E.x e. om y = suc x)
15		hbra1 1731	. . . . . . . . . . . 12 |- (A.x e. om (x e. A -> suc x e. A) -> A.xA.x e. om (x e. A -> suc x e. A))
16		ax-17 1004	. . . . . . . . . . . 12 |- ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> A.x(y e. (om \ A) /\ ((om \ A) i^i y) = (/)))
17	15, 16	hban 1042	. . . . . . . . . . 11 |- ((A.x e. om (x e. A -> suc x e. A) /\ (y e. (om \ A) /\ ((om \ A) i^i y) = (/))) -> A.x(A.x e. om (x e. A -> suc x e. A) /\ (y e. (om \ A) /\ ((om \ A) i^i y) = (/))))
18		ax-17 1004	. . . . . . . . . . 11 |- (y e. A -> A.x y e. A)
19		ra4 1738	. . . . . . . . . . . 12 |- (A.x e. om (x e. A -> suc x e. A) -> (x e. om -> (x e. A -> suc x e. A)))
20		visset 1857	. . . . . . . . . . . . . . . . . . 19 |- x e. V
21	20	sucid 3048	. . . . . . . . . . . . . . . . . 18 |- x e. suc x
22		eleq2 1576	. . . . . . . . . . . . . . . . . 18 |- (y = suc x -> (x e. y <-> x e. suc x))
23	21, 22	mpbiri 192	. . . . . . . . . . . . . . . . 17 |- (y = suc x -> x e. y)
24		eleq1 1575	. . . . . . . . . . . . . . . . . . 19 |- (y = suc x -> (y e. om <-> suc x e. om))
25		peano2b 3231	. . . . . . . . . . . . . . . . . . 19 |- (x e. om <-> suc x e. om)
26	24, 25	syl6bbr 540	. . . . . . . . . . . . . . . . . 18 |- (y = suc x -> (y e. om <-> x e. om))
27		neldif 2215	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. om /\ -. x e. (om \ A)) -> x e. A)
28		minel 2375	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. y /\ ((om \ A) i^i y) = (/)) -> -. x e. (om \ A))
29	27, 28	sylan2 453	. . . . . . . . . . . . . . . . . . 19 |- ((x e. om /\ (x e. y /\ ((om \ A) i^i y) = (/))) -> x e. A)
30	29	exp32 377	. . . . . . . . . . . . . . . . . 18 |- (x e. om -> (x e. y -> (((om \ A) i^i y) = (/) -> x e. A)))
31	26, 30	syl6bi 212	. . . . . . . . . . . . . . . . 17 |- (y = suc x -> (y e. om -> (x e. y -> (((om \ A) i^i y) = (/) -> x e. A))))
32	23, 31	mpid 47	. . . . . . . . . . . . . . . 16 |- (y = suc x -> (y e. om -> (((om \ A) i^i y) = (/) -> x e. A)))
33	32, 4	syl5 21	. . . . . . . . . . . . . . 15 |- (y = suc x -> (y e. (om \ A) -> (((om \ A) i^i y) = (/) -> x e. A)))
34	33	imp3a 359	. . . . . . . . . . . . . 14 |- (y = suc x -> ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> x e. A))
35		eleq1a 1584	. . . . . . . . . . . . . . 15 |- (suc x e. A -> (y = suc x -> y e. A))
36	35	com12 11	. . . . . . . . . . . . . 14 |- (y = suc x -> (suc x e. A -> y e. A))
37	34, 36	imim12d 29	. . . . . . . . . . . . 13 |- (y = suc x -> ((x e. A -> suc x e. A) -> ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> y e. A)))
38	37	com13 33	. . . . . . . . . . . 12 |- ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> ((x e. A -> suc x e. A) -> (y = suc x -> y e. A)))
39	19, 38	sylan9 470	. . . . . . . . . . 11 |- ((A.x e. om (x e. A -> suc x e. A) /\ (y e. (om \ A) /\ ((om \ A) i^i y) = (/))) -> (x e. om -> (y = suc x -> y e. A)))
40	17, 18, 39	r19.23ad 1789	. . . . . . . . . 10 |- ((A.x e. om (x e. A -> suc x e. A) /\ (y e. (om \ A) /\ ((om \ A) i^i y) = (/))) -> (E.x e. om y = suc x -> y e. A))
41	40	exp32 377	. . . . . . . . 9 |- (A.x e. om (x e. A -> suc x e. A) -> (y e. (om \ A) -> (((om \ A) i^i y) = (/) -> (E.x e. om y = suc x -> y e. A))))
42	41	a1i 8	. . . . . . . 8 |- ((/) e. A -> (A.x e. om (x e. A -> suc x e. A) -> (y e. (om \ A) -> (((om \ A) i^i y) = (/) -> (E.x e. om y = suc x -> y e. A)))))
43	42	imp41 366	. . . . . . 7 |- (((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) /\ ((om \ A) i^i y) = (/)) -> (E.x e. om y = suc x -> y e. A))
44	14, 43	mpd 26	. . . . . 6 |- (((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) /\ ((om \ A) i^i y) = (/)) -> y e. A)
45	44	ex 371	. . . . 5 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> (((om \ A) i^i y) = (/) -> y e. A))
46	2, 45	mtod 107	. . . 4 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> -. ((om \ A) i^i y) = (/))
47	46	nrexdv 1774	. . 3 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> -. E.y e. (om \ A)((om \ A) i^i y) = (/))
48		ordom 3225	. . . . 5 |- Ord om
49		difss 2217	. . . . 5 |- (om \ A) (_ om
50		tz7.5 2993	. . . . 5 |- ((Ord om /\ (om \ A) (_ om /\ (om \ A) =/= (/)) -> E.y e. (om \ A)((om \ A) i^i y) = (/))
51	48, 49, 50	mp3an12 909	. . . 4 |- ((om \ A) =/= (/) -> E.y e. (om \ A)((om \ A) i^i y) = (/))
52	51	necon1bi 1650	. . 3 |- (-. E.y e. (om \ A)((om \ A) i^i y) = (/) -> (om \ A) = (/))
53	47, 52	syl 10	. 2 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> (om \ A) = (/))
54		ssdif0 2378	. 2 |- (om (_ A <-> (om \ A) = (/))
55	53, 54	sylibr 198	1 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   = wceq 989   e. wcel 991   =/= wne 1626  A.wral 1689  E.wrex 1690   \ cdif 2094   i^i cin 2096   (_ wss 2097  (/)c0 2330  Ord word 2971  suc csuc 2974  omcom 3215

This theorem is referenced by:  find 3240  finds 3241  finds2 3243  omex 4763  dfom3 4767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 995  ax-gen 996  ax-8 997  ax-10 999  ax-11 1000  ax-12 1001  ax-13 1002  ax-14 1003  ax-17 1004  ax-4 1006  ax-5o 1008  ax-6o 1011  ax-9o 1156  ax-10o 1174  ax-16 1244  ax-11o 1252  ax-ext 1498  ax-sep 2773  ax-nul 2780  ax-pow 2813  ax-pr 2851  ax-un 3086

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1014  df-sb 1206  df-eu 1419  df-mo 1420  df-clab 1504  df-cleq 1509  df-clel 1512  df-ne 1628  df-ral 1693  df-rex 1694  df-v 1856  df-dif 2099  df-un 2100  df-in 2101  df-ss 2103  df-nul 2331  df-if 2414  df-pw 2454  df-sn 2465  df-pr 2466  df-tp 2468  df-op 2469  df-uni 2565  df-br 2688  df-opab 2736  df-tr 2750  df-eprel 2906  df-po 2914  df-so 2926  df-fr 2944  df-we 2959  df-ord 2975  df-on 2976  df-lim 2977  df-suc 2978  df-om 3216

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