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Theorem nnsucsssuc 6293
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4354, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4371. (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 3062 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 suceq 4253 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32sseq1d 3068 . . . . . 6 (𝑥 = 𝐴 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
41, 3imbi12d 233 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
54imbi2d 229 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)) ↔ (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))))
6 sseq1 3062 . . . . . 6 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ⊆ 𝐵))
7 suceq 4253 . . . . . . 7 (𝑥 = ∅ → suc 𝑥 = suc ∅)
87sseq1d 3068 . . . . . 6 (𝑥 = ∅ → (suc 𝑥 ⊆ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
96, 8imbi12d 233 . . . . 5 (𝑥 = ∅ → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵)))
10 sseq1 3062 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
11 suceq 4253 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1211sseq1d 3068 . . . . . 6 (𝑥 = 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝑦 ⊆ suc 𝐵))
1310, 12imbi12d 233 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)))
14 sseq1 3062 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝐵 ↔ suc 𝑦𝐵))
15 suceq 4253 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1615sseq1d 3068 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
1714, 16imbi12d 233 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵)))
18 peano3 4439 . . . . . . . . 9 (𝐵 ∈ ω → suc 𝐵 ≠ ∅)
1918neneqd 2283 . . . . . . . 8 (𝐵 ∈ ω → ¬ suc 𝐵 = ∅)
20 peano2 4438 . . . . . . . . . 10 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
21 0elnn 4460 . . . . . . . . . 10 (suc 𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2220, 21syl 14 . . . . . . . . 9 (𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2322ord 681 . . . . . . . 8 (𝐵 ∈ ω → (¬ suc 𝐵 = ∅ → ∅ ∈ suc 𝐵))
2419, 23mpd 13 . . . . . . 7 (𝐵 ∈ ω → ∅ ∈ suc 𝐵)
25 nnord 4454 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
26 ordsucim 4345 . . . . . . . 8 (Ord 𝐵 → Ord suc 𝐵)
27 0ex 3987 . . . . . . . . 9 ∅ ∈ V
28 ordelsuc 4350 . . . . . . . . 9 ((∅ ∈ V ∧ Ord suc 𝐵) → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
2927, 28mpan 416 . . . . . . . 8 (Ord suc 𝐵 → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3025, 26, 293syl 17 . . . . . . 7 (𝐵 ∈ ω → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
3124, 30mpbid 146 . . . . . 6 (𝐵 ∈ ω → suc ∅ ⊆ suc 𝐵)
3231a1d 22 . . . . 5 (𝐵 ∈ ω → (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵))
33 simp3 948 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦𝐵)
34 simp1l 970 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦 ∈ ω)
35 simp1r 971 . . . . . . . . . . . 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 4350 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ Ord 𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3834, 36, 37syl2anc 404 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3933, 38mpbird 166 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦𝐵)
40 nnsucelsuc 6292 . . . . . . . . . 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 146 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦 ∈ suc 𝐵)
43 peano2 4438 . . . . . . . . . 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 4350 . . . . . . . . 9 ((suc 𝑦 ∈ ω ∧ Ord suc 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4744, 45, 46syl2anc 404 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4842, 47mpbid 146 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc suc 𝑦 ⊆ suc 𝐵)
49483expia 1148 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))
5049exp31 357 . . . . 5 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))))
519, 13, 17, 32, 50finds2 4444 . . . 4 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)))
525, 51vtoclga 2699 . . 3 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
5352imp 123 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
54 nnon 4452 . . 3 (𝐴 ∈ ω → 𝐴 ∈ On)
55 onsucsssucr 4354 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5654, 25, 55syl2an 284 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5753, 56impbid 128 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 667  w3a 927   = wceq 1296  wcel 1445  Vcvv 2633  wss 3013  c0 3302  Ord word 4213  Oncon0 4214  suc csuc 4216  ωcom 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434
This theorem is referenced by:  nnaword  6310
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