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Theorem nnsucsssuc 6388
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4425, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4442. (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 3120 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 suceq 4324 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
32sseq1d 3126 . . . . . 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 3120 . . . . . 6 (𝑥 = ∅ → (𝑥𝐵 ↔ ∅ ⊆ 𝐵))
7 suceq 4324 . . . . . . 7 (𝑥 = ∅ → suc 𝑥 = suc ∅)
87sseq1d 3126 . . . . . 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 3120 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
11 suceq 4324 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1211sseq1d 3126 . . . . . 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 3120 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝐵 ↔ suc 𝑦𝐵))
15 suceq 4324 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1615sseq1d 3126 . . . . . 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 4510 . . . . . . . . 9 (𝐵 ∈ ω → suc 𝐵 ≠ ∅)
1918neneqd 2329 . . . . . . . 8 (𝐵 ∈ ω → ¬ suc 𝐵 = ∅)
20 peano2 4509 . . . . . . . . . 10 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
21 0elnn 4532 . . . . . . . . . 10 (suc 𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2220, 21syl 14 . . . . . . . . 9 (𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
2322ord 713 . . . . . . . 8 (𝐵 ∈ ω → (¬ suc 𝐵 = ∅ → ∅ ∈ suc 𝐵))
2419, 23mpd 13 . . . . . . 7 (𝐵 ∈ ω → ∅ ∈ suc 𝐵)
25 nnord 4525 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
26 ordsucim 4416 . . . . . . . 8 (Ord 𝐵 → Ord suc 𝐵)
27 0ex 4055 . . . . . . . . 9 ∅ ∈ V
28 ordelsuc 4421 . . . . . . . . 9 ((∅ ∈ V ∧ Ord suc 𝐵) → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
2927, 28mpan 420 . . . . . . . 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 983 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → suc 𝑦𝐵)
34 simp1l 1005 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦𝐵) → 𝑦 ∈ ω)
35 simp1r 1006 . . . . . . . . . . . 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 4421 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ Ord 𝐵) → (𝑦𝐵 ↔ suc 𝑦𝐵))
3834, 36, 37syl2anc 408 . . . . . . . . . 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 6387 . . . . . . . . . 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 4509 . . . . . . . . . 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 4421 . . . . . . . . 9 ((suc 𝑦 ∈ ω ∧ Ord suc 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
4744, 45, 46syl2anc 408 . . . . . . . 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 1183 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵)) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))
5049exp31 361 . . . . 5 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦𝐵 → suc 𝑦 ⊆ suc 𝐵) → (suc 𝑦𝐵 → suc suc 𝑦 ⊆ suc 𝐵))))
519, 13, 17, 32, 50finds2 4515 . . . 4 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥𝐵 → suc 𝑥 ⊆ suc 𝐵)))
525, 51vtoclga 2752 . . 3 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵)))
5352imp 123 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
54 nnon 4523 . . 3 (𝐴 ∈ ω → 𝐴 ∈ On)
55 onsucsssucr 4425 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
5654, 25, 55syl2an 287 . 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 697  w3a 962   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071  c0 3363  Ord word 4284  Oncon0 4285  suc csuc 4287  ωcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505
This theorem is referenced by:  nnaword  6407  ennnfonelemk  11920  ennnfonelemkh  11932
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