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Theorem ordsucunielexmid 4256
 Description: The converse of sucunielr 4236 (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 4112 . . . . . . . 8 (𝑏 ∈ On → Ord 𝑏)
2 ordtr 4115 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 ∈ On → Tr 𝑏)
4 vex 2560 . . . . . . . 8 𝑏 ∈ V
54unisuc 4150 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 127 . . . . . 6 (𝑏 ∈ On → suc 𝑏 = 𝑏)
76eleq2d 2107 . . . . 5 (𝑏 ∈ On → (𝑎 suc 𝑏𝑎𝑏))
87adantl 262 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏𝑎𝑏))
9 suceloni 4227 . . . . 5 (𝑏 ∈ On → suc 𝑏 ∈ On)
10 ordsucunielexmid.1 . . . . . 6 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
11 eleq1 2100 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
12 suceq 4139 . . . . . . . . 9 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
1312eleq1d 2106 . . . . . . . 8 (𝑥 = 𝑎 → (suc 𝑥𝑦 ↔ suc 𝑎𝑦))
1411, 13imbi12d 223 . . . . . . 7 (𝑥 = 𝑎 → ((𝑥 𝑦 → suc 𝑥𝑦) ↔ (𝑎 𝑦 → suc 𝑎𝑦)))
15 unieq 3589 . . . . . . . . 9 (𝑦 = suc 𝑏 𝑦 = suc 𝑏)
1615eleq2d 2107 . . . . . . . 8 (𝑦 = suc 𝑏 → (𝑎 𝑦𝑎 suc 𝑏))
17 eleq2 2101 . . . . . . . 8 (𝑦 = suc 𝑏 → (suc 𝑎𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
1816, 17imbi12d 223 . . . . . . 7 (𝑦 = suc 𝑏 → ((𝑎 𝑦 → suc 𝑎𝑦) ↔ (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
1914, 18rspc2va 2663 . . . . . 6 (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
2010, 19mpan2 401 . . . . 5 ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
219, 20sylan2 270 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
228, 21sylbird 159 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏))
2322rgen2a 2375 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏)
2423onsucelsucexmid 4255 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∪ cuni 3580  Tr wtr 3854  Ord word 4099  Oncon0 4100  suc csuc 4102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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