MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  peano5 Structured version   Visualization version   GIF version

Theorem peano5 7037
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 7044. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
peano5 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldifn 3716 . . . . . 6 (𝑦 ∈ (ω ∖ 𝐴) → ¬ 𝑦𝐴)
21adantl 482 . . . . 5 (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ 𝑦𝐴)
3 eldifi 3715 . . . . . . . . . 10 (𝑦 ∈ (ω ∖ 𝐴) → 𝑦 ∈ ω)
43adantl 482 . . . . . . . . 9 ((∅ ∈ 𝐴𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ∈ ω)
5 elndif 3717 . . . . . . . . . 10 (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
6 eleq1 2692 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
76biimpcd 239 . . . . . . . . . . 11 (𝑦 ∈ (ω ∖ 𝐴) → (𝑦 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
87necon3bd 2810 . . . . . . . . . 10 (𝑦 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑦 ≠ ∅))
95, 8mpan9 486 . . . . . . . . 9 ((∅ ∈ 𝐴𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ≠ ∅)
10 nnsuc 7030 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
114, 9, 10syl2anc 692 . . . . . . . 8 ((∅ ∈ 𝐴𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
1211adantlr 750 . . . . . . 7 (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
1312adantr 481 . . . . . 6 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
14 nfra1 2941 . . . . . . . . . . 11 𝑥𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)
15 nfv 1845 . . . . . . . . . . 11 𝑥(𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
1614, 15nfan 1830 . . . . . . . . . 10 𝑥(∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅))
17 nfv 1845 . . . . . . . . . 10 𝑥 𝑦𝐴
18 rsp 2929 . . . . . . . . . . 11 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
19 vex 3194 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
2019sucid 5766 . . . . . . . . . . . . . . . . 17 𝑥 ∈ suc 𝑥
21 eleq2 2693 . . . . . . . . . . . . . . . . 17 (𝑦 = suc 𝑥 → (𝑥𝑦𝑥 ∈ suc 𝑥))
2220, 21mpbiri 248 . . . . . . . . . . . . . . . 16 (𝑦 = suc 𝑥𝑥𝑦)
23 eleq1 2692 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ suc 𝑥 ∈ ω))
24 peano2b 7029 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
2523, 24syl6bbr 278 . . . . . . . . . . . . . . . . 17 (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ 𝑥 ∈ ω))
26 minel 4010 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ¬ 𝑥 ∈ (ω ∖ 𝐴))
27 neldif 3718 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ (ω ∖ 𝐴)) → 𝑥𝐴)
2826, 27sylan2 491 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ω ∧ (𝑥𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → 𝑥𝐴)
2928exp32 630 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → (𝑥𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
3025, 29syl6bi 243 . . . . . . . . . . . . . . . 16 (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (𝑥𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴))))
3122, 30mpid 44 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
323, 31syl5 34 . . . . . . . . . . . . . 14 (𝑦 = suc 𝑥 → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
3332impd 447 . . . . . . . . . . . . 13 (𝑦 = suc 𝑥 → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑥𝐴))
34 eleq1a 2699 . . . . . . . . . . . . . 14 (suc 𝑥𝐴 → (𝑦 = suc 𝑥𝑦𝐴))
3534com12 32 . . . . . . . . . . . . 13 (𝑦 = suc 𝑥 → (suc 𝑥𝐴𝑦𝐴))
3633, 35imim12d 81 . . . . . . . . . . . 12 (𝑦 = suc 𝑥 → ((𝑥𝐴 → suc 𝑥𝐴) → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦𝐴)))
3736com13 88 . . . . . . . . . . 11 ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ((𝑥𝐴 → suc 𝑥𝐴) → (𝑦 = suc 𝑥𝑦𝐴)))
3818, 37sylan9 688 . . . . . . . . . 10 ((∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (𝑥 ∈ ω → (𝑦 = suc 𝑥𝑦𝐴)))
3916, 17, 38rexlimd 3024 . . . . . . . . 9 ((∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))
4039exp32 630 . . . . . . . 8 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))))
4140a1i 11 . . . . . . 7 (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴)))))
4241imp41 618 . . . . . 6 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))
4313, 42mpd 15 . . . . 5 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦𝐴)
442, 43mtand 690 . . . 4 (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
4544nrexdv 3000 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
46 ordom 7022 . . . . 5 Ord ω
47 difss 3720 . . . . 5 (ω ∖ 𝐴) ⊆ ω
48 tz7.5 5706 . . . . 5 ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
4946, 47, 48mp3an12 1411 . . . 4 ((ω ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
5049necon1bi 2824 . . 3 (¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (ω ∖ 𝐴) = ∅)
5145, 50syl 17 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∖ 𝐴) = ∅)
52 ssdif0 3921 . 2 (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
5351, 52sylibr 224 1 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cdif 3557  cin 3559  wss 3560  c0 3896  Ord word 5684  suc csuc 5687  ωcom 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-om 7014
This theorem is referenced by:  find  7039  finds  7040  finds2  7042  omex  8485  dfom3  8489
  Copyright terms: Public domain W3C validator