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Theorem ordsucunielexmid 4309
Description: The converse of sucunielr 4289 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
Hypothesis
Ref Expression
ordsucunielexmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
Assertion
Ref Expression
ordsucunielexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordsucunielexmid
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eloni 4165 . . . . . . . 8 (𝑏 ∈ On → Ord 𝑏)
2 ordtr 4168 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 ∈ On → Tr 𝑏)
4 vex 2615 . . . . . . . 8 𝑏 ∈ V
54unisuc 4203 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 120 . . . . . 6 (𝑏 ∈ On → suc 𝑏 = 𝑏)
76eleq2d 2152 . . . . 5 (𝑏 ∈ On → (𝑎 suc 𝑏𝑎𝑏))
87adantl 271 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏𝑎𝑏))
9 suceloni 4280 . . . . 5 (𝑏 ∈ On → suc 𝑏 ∈ On)
10 ordsucunielexmid.1 . . . . . 6 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
11 eleq1 2145 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
12 suceq 4192 . . . . . . . . 9 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
1312eleq1d 2151 . . . . . . . 8 (𝑥 = 𝑎 → (suc 𝑥𝑦 ↔ suc 𝑎𝑦))
1411, 13imbi12d 232 . . . . . . 7 (𝑥 = 𝑎 → ((𝑥 𝑦 → suc 𝑥𝑦) ↔ (𝑎 𝑦 → suc 𝑎𝑦)))
15 unieq 3636 . . . . . . . . 9 (𝑦 = suc 𝑏 𝑦 = suc 𝑏)
1615eleq2d 2152 . . . . . . . 8 (𝑦 = suc 𝑏 → (𝑎 𝑦𝑎 suc 𝑏))
17 eleq2 2146 . . . . . . . 8 (𝑦 = suc 𝑏 → (suc 𝑎𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
1816, 17imbi12d 232 . . . . . . 7 (𝑦 = suc 𝑏 → ((𝑎 𝑦 → suc 𝑎𝑦) ↔ (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
1914, 18rspc2va 2724 . . . . . 6 (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
2010, 19mpan2 416 . . . . 5 ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
219, 20sylan2 280 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
228, 21sylbird 168 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏))
2322rgen2a 2423 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏)
2423onsucelsucexmid 4308 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  wral 2353   cuni 3627  Tr wtr 3901  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-in1 577  ax-in2 578  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-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  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: (None)
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