Theorem peano5 3786

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 3794.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem peano5

Step	Hyp	Ref	Expression
1		eldifn 2563	. . . . . 6 |- (y e. (om \ A) -> -. y e. A)
2	1	adantl 422	. . . . 5 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> -. y e. A)
3		nnsuc 3780	. . . . . . . . 9 |- ((y e. om /\ y =/= (/)) -> E.x e. om y = suc x)
4		eldifi 2562	. . . . . . . . . 10 |- (y e. (om \ A) -> y e. om)
5	4	adantl 422	. . . . . . . . 9 |- (((/) e. A /\ y e. (om \ A)) -> y e. om)
6		eleq1 1794	. . . . . . . . . . . . 13 |- (y = (/) -> (y e. (om \ A) <-> (/) e. (om \ A)))
7	6	biimpcd 171	. . . . . . . . . . . 12 |- (y e. (om \ A) -> (y = (/) -> (/) e. (om \ A)))
8	7	necon3bd 1874	. . . . . . . . . . 11 |- (y e. (om \ A) -> (-. (/) e. (om \ A) -> y =/= (/)))
9		elndif 2564	. . . . . . . . . . 11 |- ((/) e. A -> -. (/) e. (om \ A))
10	8, 9	syl5com 63	. . . . . . . . . 10 |- ((/) e. A -> (y e. (om \ A) -> y =/= (/)))
11	10	imp 375	. . . . . . . . 9 |- (((/) e. A /\ y e. (om \ A)) -> y =/= (/))
12	3, 5, 11	sylanc 521	. . . . . . . 8 |- (((/) e. A /\ y e. (om \ A)) -> E.x e. om y = suc x)
13	12	adantlr 427	. . . . . . 7 |- ((((/) 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 423	. . . . . 6 |- (((((/) 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 1981	. . . . . . . . . . 11 |- (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 1155	. . . . . . . . . . 11 |- ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> A.x(y e. (om \ A) /\ ((om \ A) i^i y) = (/)))
17	15, 16	hban 1194	. . . . . . . . . 10 |- ((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 1155	. . . . . . . . . 10 |- (y e. A -> A.x y e. A)
19		ra4 1989	. . . . . . . . . . 11 |- (A.x e. om (x e. A -> suc x e. A) -> (x e. om -> (x e. A -> suc x e. A)))
20		visset 2128	. . . . . . . . . . . . . . . . . 18 |- x e. _V
21	20	sucid 3558	. . . . . . . . . . . . . . . . 17 |- x e. suc x
22		eleq2 1795	. . . . . . . . . . . . . . . . 17 |- (y = suc x -> (x e. y <-> x e. suc x))
23	21, 22	mpbiri 210	. . . . . . . . . . . . . . . 16 |- (y = suc x -> x e. y)
24		eleq1 1794	. . . . . . . . . . . . . . . . . 18 |- (y = suc x -> (y e. om <-> suc x e. om))
25		peano2b 3779	. . . . . . . . . . . . . . . . . 18 |- (x e. om <-> suc x e. om)
26	24, 25	syl6bbr 594	. . . . . . . . . . . . . . . . 17 |- (y = suc x -> (y e. om <-> x e. om))
27		neldif 2565	. . . . . . . . . . . . . . . . . . 19 |- ((x e. om /\ -. x e. (om \ A)) -> x e. A)
28		minel 2753	. . . . . . . . . . . . . . . . . . 19 |- ((x e. y /\ ((om \ A) i^i y) = (/)) -> -. x e. (om \ A))
29	27, 28	sylan2 498	. . . . . . . . . . . . . . . . . 18 |- ((x e. om /\ (x e. y /\ ((om \ A) i^i y) = (/))) -> x e. A)
30	29	exp32 406	. . . . . . . . . . . . . . . . 17 |- (x e. om -> (x e. y -> (((om \ A) i^i y) = (/) -> x e. A)))
31	26, 30	syl6bi 230	. . . . . . . . . . . . . . . 16 |- (y = suc x -> (y e. om -> (x e. y -> (((om \ A) i^i y) = (/) -> x e. A))))
32	23, 31	mpid 58	. . . . . . . . . . . . . . 15 |- (y = suc x -> (y e. om -> (((om \ A) i^i y) = (/) -> x e. A)))
33	32, 4	syl5 20	. . . . . . . . . . . . . 14 |- (y = suc x -> (y e. (om \ A) -> (((om \ A) i^i y) = (/) -> x e. A)))
34	33	imp3a 386	. . . . . . . . . . . . 13 |- (y = suc x -> ((y e. (om \ A) /\ ((om \ A) i^i y) = (/)) -> x e. A))
35		eleq1a 1803	. . . . . . . . . . . . . 14 |- (suc x e. A -> (y = suc x -> y e. A))
36	35	com12 14	. . . . . . . . . . . . 13 |- (y = suc x -> (suc x e. A -> y e. A))
37	34, 36	imim12d 69	. . . . . . . . . . . 12 |- (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 37	. . . . . . . . . . 11 |- ((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 515	. . . . . . . . . 10 |- ((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 2047	. . . . . . . . 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))
41	40	exp32 406	. . . . . . . 8 |- (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	. . . . . . 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)))))
43	42	imp41 393	. . . . . 6 |- (((((/) 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 29	. . . . 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)
45	2, 44	mtand 518	. . . 4 |- ((((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) /\ y e. (om \ A)) -> -. ((om \ A) i^i y) = (/))
46	45	nrexdv 2027	. . 3 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> -. E.y e. (om \ A)((om \ A) i^i y) = (/))
47		ordom 3771	. . . . 5 |- Ord om
48		difss 2567	. . . . 5 |- (om \ A) C_ om
49		tz7.5 3494	. . . . 5 |- ((Ord om /\ (om \ A) C_ om /\ (om \ A) =/= (/)) -> E.y e. (om \ A)((om \ A) i^i y) = (/))
50	47, 48, 49	mp3an12 1028	. . . 4 |- ((om \ A) =/= (/) -> E.y e. (om \ A)((om \ A) i^i y) = (/))
51	50	necon1bi 1883	. . 3 |- (-. E.y e. (om \ A)((om \ A) i^i y) = (/) -> (om \ A) = (/))
52	46, 51	syl 12	. 2 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> (om \ A) = (/))
53		ssdif0 2758	. 2 |- (om C_ A <-> (om \ A) = (/))
54	52, 53	sylibr 216	1 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 239   = wceq 1136   e. wcel 1138   =/= wne 1854  A.wral 1939  E.wrex 1940   \ cdif 2423   i^i cin 2425   C_ wss 2426  (/)c0 2701  Ord word 3471  suc csuc 3474  omcom 3760

This theorem is referenced by:  find 3788  findOLD 3789  finds 3790  finds2 3792  omex 5542  dfom3 5546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-ral 1943  df-rex 1944  df-rab 1946  df-v 2127  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761

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