Theorem peano5 6723

Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five 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 step, is derived from this theorem as theorem findes 6730. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
peano5

Distinct variable group:   ,

Proof of Theorem peano5

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifn 3626	. . . . . 6
2	1	adantl 466	. . . . 5
3		eldifi 3625	. . . . . . . . . 10
4	3	adantl 466	. . . . . . . . 9
5		elndif 3627	. . . . . . . . . 10
6		eleq1 2529	. . . . . . . . . . . 12
7	6	biimpcd 224	. . . . . . . . . . 11
8	7	necon3bd 2669	. . . . . . . . . 10
9	5, 8	mpan9 469	. . . . . . . . 9
10		nnsuc 6717	. . . . . . . . 9
11	4, 9, 10	syl2anc 661	. . . . . . . 8
12	11	adantlr 714	. . . . . . 7
13	12	adantr 465	. . . . . 6
14		nfra1 2838	. . . . . . . . . . 11
15		nfv 1707	. . . . . . . . . . 11
16	14, 15	nfan 1928	. . . . . . . . . 10
17		nfv 1707	. . . . . . . . . 10
18		rsp 2823	. . . . . . . . . . 11
19		vex 3112	. . . . . . . . . . . . . . . . . 18
20	19	sucid 4962	. . . . . . . . . . . . . . . . 17
21		eleq2 2530	. . . . . . . . . . . . . . . . 17
22	20, 21	mpbiri 233	. . . . . . . . . . . . . . . 16
23		eleq1 2529	. . . . . . . . . . . . . . . . . 18
24		peano2b 6716	. . . . . . . . . . . . . . . . . 18
25	23, 24	syl6bbr 263	. . . . . . . . . . . . . . . . 17
26		minel 3882	. . . . . . . . . . . . . . . . . . 19
27		neldif 3628	. . . . . . . . . . . . . . . . . . 19
28	26, 27	sylan2 474	. . . . . . . . . . . . . . . . . 18
29	28	exp32 605	. . . . . . . . . . . . . . . . 17
30	25, 29	syl6bi 228	. . . . . . . . . . . . . . . 16
31	22, 30	mpid 41	. . . . . . . . . . . . . . 15
32	3, 31	syl5 32	. . . . . . . . . . . . . 14
33	32	impd 431	. . . . . . . . . . . . 13
34		eleq1a 2540	. . . . . . . . . . . . . 14
35	34	com12 31	. . . . . . . . . . . . 13
36	33, 35	imim12d 74	. . . . . . . . . . . 12
37	36	com13 80	. . . . . . . . . . 11
38	18, 37	sylan9 657	. . . . . . . . . 10
39	16, 17, 38	rexlimd 2941	. . . . . . . . 9
40	39	exp32 605	. . . . . . . 8
41	40	a1i 11	. . . . . . 7
42	41	imp41 593	. . . . . 6
43	13, 42	mpd 15	. . . . 5
44	2, 43	mtand 659	. . . 4
45	44	nrexdv 2913	. . 3
46		ordom 6709	. . . . 5
47		difss 3630	. . . . 5
48		tz7.5 4904	. . . . 5
49	46, 47, 48	mp3an12 1314	. . . 4
50	49	necon1bi 2690	. . 3
51	45, 50	syl 16	. 2
52		ssdif0 3885	. 2
53	51, 52	sylibr 212	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  \cdif 3472  i^icin 3474  C_wss 3475  c0 3784  Ordword 4882  succsuc 4885  com 6700

This theorem is referenced by:  find  6725  finds  6726  finds2  6728  omex  8081  dfom3  8085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-om 6701