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Theorem nnsucelsuc 6492
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4508, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4530. (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 2241 . . . 4 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 suceq 4403 . . . . 5 (𝑥 = ∅ → suc 𝑥 = suc ∅)
32eleq2d 2247 . . . 4 (𝑥 = ∅ → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc ∅))
41, 3imbi12d 234 . . 3 (𝑥 = ∅ → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)))
5 eleq2 2241 . . . 4 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 suceq 4403 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
76eleq2d 2247 . . . 4 (𝑥 = 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝑦))
85, 7imbi12d 234 . . 3 (𝑥 = 𝑦 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴𝑦 → suc 𝐴 ∈ suc 𝑦)))
9 eleq2 2241 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 suceq 4403 . . . . 5 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1110eleq2d 2247 . . . 4 (𝑥 = suc 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc suc 𝑦))
129, 11imbi12d 234 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
13 eleq2 2241 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 suceq 4403 . . . . 5 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
1514eleq2d 2247 . . . 4 (𝑥 = 𝐵 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝐵))
1613, 15imbi12d 234 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵)))
17 noel 3427 . . . 4 ¬ 𝐴 ∈ ∅
1817pm2.21i 646 . . 3 (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19 elsuci 4404 . . . . . . . 8 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
2019adantl 277 . . . . . . 7 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴𝑦𝐴 = 𝑦))
21 simpl 109 . . . . . . . 8 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴𝑦 → suc 𝐴 ∈ suc 𝑦))
22 suceq 4403 . . . . . . . . 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 786 . . . . . . 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 2741 . . . . . . . 8 𝑦 ∈ V
2726sucex 4499 . . . . . . 7 suc 𝑦 ∈ V
2827elsuc2 4408 . . . . . 6 (suc 𝐴 ∈ suc suc 𝑦 ↔ (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
2925, 28sylibr 134 . . . . 5 (((𝐴𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → suc 𝐴 ∈ suc suc 𝑦)
3029ex 115 . . . 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 4600 . 2 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵))
33 nnon 4610 . . 3 (𝐵 ∈ ω → 𝐵 ∈ On)
34 onsucelsucr 4508 . . 3 (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
3533, 34syl 14 . 2 (𝐵 ∈ ω → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
3632, 35impbid 129 1 (𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  c0 3423  Oncon0 4364  suc csuc 4366  ωcom 4590
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591
This theorem is referenced by:  nnsucsssuc  6493  nntri3or  6494  nnsucuniel  6496  nnaordi  6509  ennnfonelemhom  12416
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