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Theorem nnsucelsuc 6495
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4509, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4531. (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 4404 . . . . 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 4404 . . . . 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 4404 . . . . 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 4404 . . . . 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 3428 . . . 4 ¬ 𝐴 ∈ ∅
1817pm2.21i 646 . . 3 (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19 elsuci 4405 . . . . . . . 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 4404 . . . . . . . . 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 2742 . . . . . . . 8 𝑦 ∈ V
2726sucex 4500 . . . . . . 7 suc 𝑦 ∈ V
2827elsuc2 4409 . . . . . 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 4601 . 2 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴 ∈ suc 𝐵))
33 nnon 4611 . . 3 (𝐵 ∈ ω → 𝐵 ∈ On)
34 onsucelsucr 4509 . . 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 3424  Oncon0 4365  suc csuc 4367  ωcom 4591
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589
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 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592
This theorem is referenced by:  nnsucsssuc  6496  nntri3or  6497  nnsucuniel  6499  nnaordi  6512  ennnfonelemhom  12419
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