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Theorem nnsucsssuc 6518
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4526, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4544. (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 3193 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 suceq 4420 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32sseq1d 3199 . . . . . 6 (𝑥 = 𝐴 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
41, 3imbi12d 234 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
54imbi2d 230 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)) ↔ (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))))
6 sseq1 3193 . . . . . 6 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ⊆ 𝐵))
7 suceq 4420 . . . . . . 7 (𝑥 = ∅ → suc 𝑥 = suc ∅)
87sseq1d 3199 . . . . . 6 (𝑥 = ∅ → (suc 𝑥 ⊆ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
96, 8imbi12d 234 . . . . 5 (𝑥 = ∅ → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵)))
10 sseq1 3193 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
11 suceq 4420 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1211sseq1d 3199 . . . . . 6 (𝑥 = 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝑦 ⊆ suc 𝐵))
1310, 12imbi12d 234 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)))
14 sseq1 3193 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝐵 ↔ suc 𝑦𝐵))
15 suceq 4420 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1615sseq1d 3199 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
1714, 16imbi12d 234 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵)))
18 peano3 4613 . . . . . . . . 9 (𝐵 ∈ ω → suc 𝐵 ≠ ∅)
1918neneqd 2381 . . . . . . . 8 (𝐵 ∈ ω → ¬ suc 𝐵 = ∅)
20 peano2 4612 . . . . . . . . . 10 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
21 0elnn 4636 . . . . . . . . . 10 (suc 𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2220, 21syl 14 . . . . . . . . 9 (𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2322ord 725 . . . . . . . 8 (𝐵 ∈ ω → (¬ suc 𝐵 = ∅ → ∅ ∈ suc 𝐵))
2419, 23mpd 13 . . . . . . 7 (𝐵 ∈ ω → ∅ ∈ suc 𝐵)
25 nnord 4629 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
26 ordsucim 4517 . . . . . . . 8 (Ord 𝐵 → Ord suc 𝐵)
27 0ex 4145 . . . . . . . . 9 ∅ ∈ V
28 ordelsuc 4522 . . . . . . . . 9 ((∅ ∈ V ∧ Ord suc 𝐵) → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
2927, 28mpan 424 . . . . . . . 8 (Ord suc 𝐵 → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3025, 26, 293syl 17 . . . . . . 7 (𝐵 ∈ ω → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3124, 30mpbid 147 . . . . . 6 (𝐵 ∈ ω → suc ∅ ⊆ suc 𝐵)
3231a1d 22 . . . . 5 (𝐵 ∈ ω → (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵))
33 simp3 1001 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦𝐵)
34 simp1l 1023 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦 ∈ ω)
35 simp1r 1024 . . . . . . . . . . . 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 4522 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ Ord 𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3834, 36, 37syl2anc 411 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3933, 38mpbird 167 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦𝐵)
40 nnsucelsuc 6517 . . . . . . . . . 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 147 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦 ∈ suc 𝐵)
43 peano2 4612 . . . . . . . . . 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 4522 . . . . . . . . 9 ((suc 𝑦 ∈ ω ∧ Ord suc 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4744, 45, 46syl2anc 411 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4842, 47mpbid 147 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc suc 𝑦 ⊆ suc 𝐵)
49483expia 1207 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))
5049exp31 364 . . . . 5 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))))
519, 13, 17, 32, 50finds2 4618 . . . 4 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)))
525, 51vtoclga 2818 . . 3 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
5352imp 124 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
54 nnon 4627 . . 3 (𝐴 ∈ ω → 𝐴 ∈ On)
55 onsucsssucr 4526 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5654, 25, 55syl2an 289 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5753, 56impbid 129 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  w3a 980   = wceq 1364  wcel 2160  Vcvv 2752  wss 3144  c0 3437  Ord word 4380  Oncon0 4381  suc csuc 4383  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608
This theorem is referenced by:  nnaword  6537  ennnfonelemk  12454  ennnfonelemkh  12466
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