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Theorem ordsucunielexmid 4457
 Description: The converse of sucunielr 4437 (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 4308 . . . . . . . 8 (𝑏 ∈ On → Ord 𝑏)
2 ordtr 4311 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 ∈ On → Tr 𝑏)
4 vex 2694 . . . . . . . 8 𝑏 ∈ V
54unisuc 4346 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 121 . . . . . 6 (𝑏 ∈ On → suc 𝑏 = 𝑏)
76eleq2d 2211 . . . . 5 (𝑏 ∈ On → (𝑎 suc 𝑏𝑎𝑏))
87adantl 275 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏𝑎𝑏))
9 suceloni 4428 . . . . 5 (𝑏 ∈ On → suc 𝑏 ∈ On)
10 ordsucunielexmid.1 . . . . . 6 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
11 eleq1 2204 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
12 suceq 4335 . . . . . . . . 9 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
1312eleq1d 2210 . . . . . . . 8 (𝑥 = 𝑎 → (suc 𝑥𝑦 ↔ suc 𝑎𝑦))
1411, 13imbi12d 233 . . . . . . 7 (𝑥 = 𝑎 → ((𝑥 𝑦 → suc 𝑥𝑦) ↔ (𝑎 𝑦 → suc 𝑎𝑦)))
15 unieq 3755 . . . . . . . . 9 (𝑦 = suc 𝑏 𝑦 = suc 𝑏)
1615eleq2d 2211 . . . . . . . 8 (𝑦 = suc 𝑏 → (𝑎 𝑦𝑎 suc 𝑏))
17 eleq2 2205 . . . . . . . 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 2809 . . . . . 6 (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
2010, 19mpan2 422 . . . . 5 ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
219, 20sylan2 284 . . . 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 2491 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏)
2423onsucelsucexmid 4456 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 2112  ∀wral 2418  ∪ cuni 3746  Tr wtr 4036  Ord word 4295  Oncon0 4296  suc csuc 4298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-nul 4064  ax-pow 4108  ax-pr 4142  ax-un 4366 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540  df-pr 3541  df-uni 3747  df-tr 4037  df-iord 4299  df-on 4301  df-suc 4304 This theorem is referenced by: (None)
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