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Theorem ordsucunielexmid 4384
Description: The converse of sucunielr 4364 (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 4235 . . . . . . . 8 (𝑏 ∈ On → Ord 𝑏)
2 ordtr 4238 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 ∈ On → Tr 𝑏)
4 vex 2644 . . . . . . . 8 𝑏 ∈ V
54unisuc 4273 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 121 . . . . . 6 (𝑏 ∈ On → suc 𝑏 = 𝑏)
76eleq2d 2169 . . . . 5 (𝑏 ∈ On → (𝑎 suc 𝑏𝑎𝑏))
87adantl 273 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏𝑎𝑏))
9 suceloni 4355 . . . . 5 (𝑏 ∈ On → suc 𝑏 ∈ On)
10 ordsucunielexmid.1 . . . . . 6 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
11 eleq1 2162 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
12 suceq 4262 . . . . . . . . 9 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
1312eleq1d 2168 . . . . . . . 8 (𝑥 = 𝑎 → (suc 𝑥𝑦 ↔ suc 𝑎𝑦))
1411, 13imbi12d 233 . . . . . . 7 (𝑥 = 𝑎 → ((𝑥 𝑦 → suc 𝑥𝑦) ↔ (𝑎 𝑦 → suc 𝑎𝑦)))
15 unieq 3692 . . . . . . . . 9 (𝑦 = suc 𝑏 𝑦 = suc 𝑏)
1615eleq2d 2169 . . . . . . . 8 (𝑦 = suc 𝑏 → (𝑎 𝑦𝑎 suc 𝑏))
17 eleq2 2163 . . . . . . . 8 (𝑦 = suc 𝑏 → (suc 𝑎𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
1816, 17imbi12d 233 . . . . . . 7 (𝑦 = suc 𝑏 → ((𝑎 𝑦 → suc 𝑎𝑦) ↔ (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
1914, 18rspc2va 2757 . . . . . 6 (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
2010, 19mpan2 419 . . . . 5 ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
219, 20sylan2 282 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
228, 21sylbird 169 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏))
2322rgen2a 2445 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏)
2423onsucelsucexmid 4383 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 670   = wceq 1299  wcel 1448  wral 2375   cuni 3683  Tr wtr 3966  Ord word 4222  Oncon0 4223  suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231
This theorem is referenced by: (None)
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