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Theorem nnsucelsuc 6266
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4338, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4359. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nnsucelsuc (𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Proof of Theorem nnsucelsuc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2152 . . . 4 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 suceq 4238 . . . . 5 (𝑥 = ∅ → suc 𝑥 = suc ∅)
32eleq2d 2158 . . . 4 (𝑥 = ∅ → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc ∅))
41, 3imbi12d 233 . . 3 (𝑥 = ∅ → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)))
5 eleq2 2152 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 suceq 4238 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
76eleq2d 2158 . . . 4 (𝑥 = 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝑦))
85, 7imbi12d 233 . . 3 (𝑥 = 𝑦 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴𝑦 → suc 𝐴 ∈ suc 𝑦)))
9 eleq2 2152 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 suceq 4238 . . . . 5 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1110eleq2d 2158 . . . 4 (𝑥 = suc 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc suc 𝑦))
129, 11imbi12d 233 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
13 eleq2 2152 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 suceq 4238 . . . . 5 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
1514eleq2d 2158 . . . 4 (𝑥 = 𝐵 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝐵))
1613, 15imbi12d 233 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵)))
17 noel 3291 . . . 4 ¬ 𝐴 ∈ ∅
1817pm2.21i 611 . . 3 (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19 elsuci 4239 . . . . . . . 8 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
2019adantl 272 . . . . . . 7 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴𝑦𝐴 = 𝑦))
21 simpl 108 . . . . . . . 8 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴𝑦 → suc 𝐴 ∈ suc 𝑦))
22 suceq 4238 . . . . . . . . 9 (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
2322a1i 9 . . . . . . . 8 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦))
2421, 23orim12d 736 . . . . . . 7 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → ((𝐴𝑦𝐴 = 𝑦) → (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦)))
2520, 24mpd 13 . . . . . 6 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
26 vex 2623 . . . . . . . 8 𝑦 ∈ V
2726sucex 4329 . . . . . . 7 suc 𝑦 ∈ V
2827elsuc2 4243 . . . . . 6 (suc 𝐴 ∈ suc suc 𝑦 ↔ (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
2925, 28sylibr 133 . . . . 5 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → suc 𝐴 ∈ suc suc 𝑦)
3029ex 114 . . . 4 ((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) → (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦))
3130a1i 9 . . 3 (𝑦 ∈ ω → ((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) → (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
324, 8, 12, 16, 18, 31finds 4428 . 2 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵))
33 nnon 4437 . . 3 (𝐵 ∈ ω → 𝐵 ∈ On)
34 onsucelsucr 4338 . . 3 (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
3533, 34syl 14 . 2 (𝐵 ∈ ω → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
3632, 35impbid 128 1 (𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 665   = wceq 1290  wcel 1439  c0 3287  Oncon0 4199  suc csuc 4201  ωcom 4418
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419
This theorem is referenced by:  nnsucsssuc  6267  nntri3or  6268  nnsucuniel  6270  nnaordi  6281
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