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Theorem peano5OLD 7898
Description: Obsolete version of peano5 7897 as of 3-Oct-2024. (Contributed by NM, 18-Feb-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
peano5OLD ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldifn 4120 . . . . . 6 (𝑦 ∈ (ω ∖ 𝐴) → ¬ 𝑦𝐴)
21adantl 480 . . . . 5 (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ 𝑦𝐴)
3 eldifi 4119 . . . . . . . 8 (𝑦 ∈ (ω ∖ 𝐴) → 𝑦 ∈ ω)
4 elndif 4121 . . . . . . . . 9 (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
5 eleq1 2813 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
65biimpcd 248 . . . . . . . . . 10 (𝑦 ∈ (ω ∖ 𝐴) → (𝑦 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
76necon3bd 2944 . . . . . . . . 9 (𝑦 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑦 ≠ ∅))
84, 7mpan9 505 . . . . . . . 8 ((∅ ∈ 𝐴𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ≠ ∅)
9 nnsuc 7886 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
103, 8, 9syl2an2 684 . . . . . . 7 ((∅ ∈ 𝐴𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
1110ad4ant13 749 . . . . . 6 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
12 nfra1 3272 . . . . . . . . . . 11 𝑥𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)
13 nfv 1909 . . . . . . . . . . 11 𝑥(𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
1412, 13nfan 1894 . . . . . . . . . 10 𝑥(∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅))
15 nfv 1909 . . . . . . . . . 10 𝑥 𝑦𝐴
16 rsp 3235 . . . . . . . . . . 11 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
17 vex 3467 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
1817sucid 6446 . . . . . . . . . . . . . . . . 17 𝑥 ∈ suc 𝑥
19 eleq2 2814 . . . . . . . . . . . . . . . . 17 (𝑦 = suc 𝑥 → (𝑥𝑦𝑥 ∈ suc 𝑥))
2018, 19mpbiri 257 . . . . . . . . . . . . . . . 16 (𝑦 = suc 𝑥𝑥𝑦)
21 eleq1 2813 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ suc 𝑥 ∈ ω))
22 peano2b 7885 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
2321, 22bitr4di 288 . . . . . . . . . . . . . . . . 17 (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ 𝑥 ∈ ω))
24 minel 4461 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ¬ 𝑥 ∈ (ω ∖ 𝐴))
25 neldif 4122 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ (ω ∖ 𝐴)) → 𝑥𝐴)
2624, 25sylan2 591 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ω ∧ (𝑥𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → 𝑥𝐴)
2726exp32 419 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → (𝑥𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
2823, 27biimtrdi 252 . . . . . . . . . . . . . . . 16 (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (𝑥𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴))))
2920, 28mpid 44 . . . . . . . . . . . . . . 15 (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
303, 29syl5 34 . . . . . . . . . . . . . 14 (𝑦 = suc 𝑥 → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥𝐴)))
3130impd 409 . . . . . . . . . . . . 13 (𝑦 = suc 𝑥 → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑥𝐴))
32 eleq1a 2820 . . . . . . . . . . . . . 14 (suc 𝑥𝐴 → (𝑦 = suc 𝑥𝑦𝐴))
3332com12 32 . . . . . . . . . . . . 13 (𝑦 = suc 𝑥 → (suc 𝑥𝐴𝑦𝐴))
3431, 33imim12d 81 . . . . . . . . . . . 12 (𝑦 = suc 𝑥 → ((𝑥𝐴 → suc 𝑥𝐴) → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦𝐴)))
3534com13 88 . . . . . . . . . . 11 ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ((𝑥𝐴 → suc 𝑥𝐴) → (𝑦 = suc 𝑥𝑦𝐴)))
3616, 35sylan9 506 . . . . . . . . . 10 ((∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (𝑥 ∈ ω → (𝑦 = suc 𝑥𝑦𝐴)))
3714, 15, 36rexlimd 3254 . . . . . . . . 9 ((∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))
3837exp32 419 . . . . . . . 8 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))))
3938a1i 11 . . . . . . 7 (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴)))))
4039imp41 424 . . . . . 6 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥𝑦𝐴))
4111, 40mpd 15 . . . . 5 ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦𝐴)
422, 41mtand 814 . . . 4 (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
4342nrexdv 3139 . . 3 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
44 ordom 7878 . . . . 5 Ord ω
45 difss 4124 . . . . 5 (ω ∖ 𝐴) ⊆ ω
46 tz7.5 6385 . . . . 5 ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
4744, 45, 46mp3an12 1447 . . . 4 ((ω ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
4847necon1bi 2959 . . 3 (¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (ω ∖ 𝐴) = ∅)
4943, 48syl 17 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∖ 𝐴) = ∅)
50 ssdif0 4359 . 2 (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
5149, 50sylibr 233 1 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2930  wral 3051  wrex 3060  cdif 3936  cin 3938  wss 3939  c0 4318  Ord word 6363  suc csuc 6366  ωcom 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7869
This theorem is referenced by: (None)
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