ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnsucsssuc GIF version

Theorem nnsucsssuc 6188
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4292, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4309. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nnsucsssuc ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem nnsucsssuc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3033 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 suceq 4196 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32sseq1d 3039 . . . . . 6 (𝑥 = 𝐴 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
41, 3imbi12d 232 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
54imbi2d 228 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)) ↔ (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))))
6 sseq1 3033 . . . . . 6 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ⊆ 𝐵))
7 suceq 4196 . . . . . . 7 (𝑥 = ∅ → suc 𝑥 = suc ∅)
87sseq1d 3039 . . . . . 6 (𝑥 = ∅ → (suc 𝑥 ⊆ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
96, 8imbi12d 232 . . . . 5 (𝑥 = ∅ → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵)))
10 sseq1 3033 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
11 suceq 4196 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1211sseq1d 3039 . . . . . 6 (𝑥 = 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝑦 ⊆ suc 𝐵))
1310, 12imbi12d 232 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)))
14 sseq1 3033 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝐵 ↔ suc 𝑦𝐵))
15 suceq 4196 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1615sseq1d 3039 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
1714, 16imbi12d 232 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵)))
18 peano3 4377 . . . . . . . . 9 (𝐵 ∈ ω → suc 𝐵 ≠ ∅)
1918neneqd 2272 . . . . . . . 8 (𝐵 ∈ ω → ¬ suc 𝐵 = ∅)
20 peano2 4376 . . . . . . . . . 10 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
21 0elnn 4398 . . . . . . . . . 10 (suc 𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2220, 21syl 14 . . . . . . . . 9 (𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2322ord 676 . . . . . . . 8 (𝐵 ∈ ω → (¬ suc 𝐵 = ∅ → ∅ ∈ suc 𝐵))
2419, 23mpd 13 . . . . . . 7 (𝐵 ∈ ω → ∅ ∈ suc 𝐵)
25 nnord 4392 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
26 ordsucim 4283 . . . . . . . 8 (Ord 𝐵 → Ord suc 𝐵)
27 0ex 3934 . . . . . . . . 9 ∅ ∈ V
28 ordelsuc 4288 . . . . . . . . 9 ((∅ ∈ V ∧ Ord suc 𝐵) → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
2927, 28mpan 415 . . . . . . . 8 (Ord suc 𝐵 → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3025, 26, 293syl 17 . . . . . . 7 (𝐵 ∈ ω → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3124, 30mpbid 145 . . . . . 6 (𝐵 ∈ ω → suc ∅ ⊆ suc 𝐵)
3231a1d 22 . . . . 5 (𝐵 ∈ ω → (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵))
33 simp3 943 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦𝐵)
34 simp1l 965 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦 ∈ ω)
35 simp1r 966 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝐵 ∈ ω)
3635, 25syl 14 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → Ord 𝐵)
37 ordelsuc 4288 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ Ord 𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3834, 36, 37syl2anc 403 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3933, 38mpbird 165 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦𝐵)
40 nnsucelsuc 6187 . . . . . . . . . 10 (𝐵 ∈ ω → (𝑦𝐵 ↔ suc 𝑦 ∈ suc 𝐵))
4135, 40syl 14 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (𝑦𝐵 ↔ suc 𝑦 ∈ suc 𝐵))
4239, 41mpbid 145 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦 ∈ suc 𝐵)
43 peano2 4376 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
4434, 43syl 14 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦 ∈ ω)
4536, 26syl 14 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → Ord suc 𝐵)
46 ordelsuc 4288 . . . . . . . . 9 ((suc 𝑦 ∈ ω ∧ Ord suc 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4744, 45, 46syl2anc 403 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4842, 47mpbid 145 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc suc 𝑦 ⊆ suc 𝐵)
49483expia 1143 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))
5049exp31 356 . . . . 5 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))))
519, 13, 17, 32, 50finds2 4382 . . . 4 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)))
525, 51vtoclga 2677 . . 3 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
5352imp 122 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
54 nnon 4390 . . 3 (𝐴 ∈ ω → 𝐴 ∈ On)
55 onsucsssucr 4292 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5654, 25, 55syl2an 283 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5753, 56impbid 127 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 922   = wceq 1287  wcel 1436  Vcvv 2614  wss 2986  c0 3272  Ord word 4156  Oncon0 4157  suc csuc 4159  ωcom 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-int 3666  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372
This theorem is referenced by:  nnaword  6203
  Copyright terms: Public domain W3C validator