Theorem peano5 6793

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 6800. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
peano5	⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifn 3581	. . . . . 6 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
2	1	adantl 475	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
3		eldifi 3580	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → 𝑦 ∈ ω)
4	3	adantl 475	. . . . . . . . 9 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ∈ ω)
5		elndif 3582	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
6		eleq1 2571	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
7	6	biimpcd 234	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (𝑦 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
8	7	necon3bd 2691	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑦 ≠ ∅))
9	5, 8	mpan9 479	. . . . . . . . 9 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ≠ ∅)
10		nnsuc 6786	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
11	4, 9, 10	syl2anc 682	. . . . . . . 8 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
12	11	adantlr 738	. . . . . . 7 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
13	12	adantr 474	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
14		nfra1 2820	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)
15		nfv 1792	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
16	14, 15	nfan 2064	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅))
17		nfv 1792	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
18		rsp 2808	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
19		vex 3069	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
20	19	sucid 5553	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ suc 𝑥
21		eleq2 2572	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ suc 𝑥))
22	20, 21	mpbiri 243	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → 𝑥 ∈ 𝑦)
23		eleq1 2571	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ suc 𝑥 ∈ ω))
24		peano2b 6785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
25	23, 24	syl6bbr 273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ 𝑥 ∈ ω))
26		minel 3860	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ¬ 𝑥 ∈ (ω ∖ 𝐴))
27		neldif 3583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ (ω ∖ 𝐴)) → 𝑥 ∈ 𝐴)
28	26, 27	sylan2 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ω ∧ (𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐴)
29	28	exp32 622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
30	25, 29	syl6bi 238	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴))))
31	22, 30	mpid 42	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
32	3, 31	syl5 33	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
33	32	impd 440	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑥 ∈ 𝐴))
34		eleq1a 2578	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ 𝐴 → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → (suc 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
36	33, 35	imim12d 78	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑥 → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)))
37	36	com13 84	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
38	18, 37	sylan9 678	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (𝑥 ∈ ω → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
39	16, 17, 38	rexlimd 2901	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
40	39	exp32 622	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))))
41	40	a1i 11	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))))
42	41	imp41 610	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
43	13, 42	mpd 15	. . . . 5 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)
44	2, 43	mtand 680	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
45	44	nrexdv 2877	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
46		ordom 6778	. . . . 5 ⊢ Ord ω
47		difss 3585	. . . . 5 ⊢ (ω ∖ 𝐴) ⊆ ω
48		tz7.5 5495	. . . . 5 ⊢ ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
49	46, 47, 48	mp3an12 1396	. . . 4 ⊢ ((ω ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
50	49	necon1bi 2705	. . 3 ⊢ (¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (ω ∖ 𝐴) = ∅)
51	45, 50	syl 17	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∖ 𝐴) = ∅)
52		ssdif0 3776	. 2 ⊢ (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
53	51, 52	sylibr 219	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 378   = wceq 1468   ∈ wcel 1937   ≠ wne 2675  ∀wral 2791  ∃wrex 2792   ∖ cdif 3423   ∩ cin 3425   ⊆ wss 3426  ∅c0 3757  Ord word 5473  suc csuc 5476  ωcom 6769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pr 4680  ax-un 6659

This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-sbc 3292  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-tr 4531  df-eprel 4791  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-om 6770

This theorem is referenced by:  find  6795  finds  6796  finds2  6798  omex  8233  dfom3  8237