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Theorem onsucelsucr 4287
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4308. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6183. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2621 . . . 4 (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2 sucexb 4276 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 132 . . 3 (suc 𝐴 ∈ suc 𝐵𝐴 ∈ V)
4 onelss 4177 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴𝐵 → suc 𝐴𝐵))
5 eqimss 3062 . . . . . . . 8 (suc 𝐴 = 𝐵 → suc 𝐴𝐵)
65a1i 9 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴𝐵))
74, 6jaod 670 . . . . . 6 (𝐵 ∈ On → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
87adantl 271 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
9 elsucg 4194 . . . . . . 7 (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
102, 9sylbi 119 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
1110adantr 270 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
12 eloni 4165 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
13 ordelsuc 4284 . . . . . 6 ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
1412, 13sylan2 280 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ suc 𝐴𝐵))
158, 11, 143imtr4d 201 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
1615impancom 256 . . 3 ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴𝐵))
173, 16mpancom 413 . 2 (suc 𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴𝐵))
1817com12 30 1 (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  Vcvv 2612  wss 2984  Ord word 4152  Oncon0 4153  suc csuc 4155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by:  nnsucelsuc  6183
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