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Theorem nnsucelsuc 6380
 Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4419, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4440. (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 2201 . . . 4 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 suceq 4319 . . . . 5 (𝑥 = ∅ → suc 𝑥 = suc ∅)
32eleq2d 2207 . . . 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 2201 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 suceq 4319 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
76eleq2d 2207 . . . 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 2201 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 suceq 4319 . . . . 5 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1110eleq2d 2207 . . . 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 2201 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 suceq 4319 . . . . 5 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
1514eleq2d 2207 . . . 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 3362 . . . 4 ¬ 𝐴 ∈ ∅
1817pm2.21i 635 . . 3 (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19 elsuci 4320 . . . . . . . 8 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
2019adantl 275 . . . . . . 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 4319 . . . . . . . . 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 775 . . . . . . 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 2684 . . . . . . . 8 𝑦 ∈ V
2726sucex 4410 . . . . . . 7 suc 𝑦 ∈ V
2827elsuc2 4324 . . . . . 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 4509 . 2 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵))
33 nnon 4518 . . 3 (𝐵 ∈ ω → 𝐵 ∈ On)
34 onsucelsucr 4419 . . 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 697   = wceq 1331   ∈ wcel 1480  ∅c0 3358  Oncon0 4280  suc csuc 4282  ωcom 4499 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 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500 This theorem is referenced by:  nnsucsssuc  6381  nntri3or  6382  nnsucuniel  6384  nnaordi  6397  ennnfonelemhom  11917
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