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Theorem ordsucunielexmid 4531
Description: The converse of sucunielr 4510 (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 4376 . . . . . . . 8 (𝑏 ∈ On → Ord 𝑏)
2 ordtr 4379 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 ∈ On → Tr 𝑏)
4 vex 2741 . . . . . . . 8 𝑏 ∈ V
54unisuc 4414 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 122 . . . . . 6 (𝑏 ∈ On → suc 𝑏 = 𝑏)
76eleq2d 2247 . . . . 5 (𝑏 ∈ On → (𝑎 suc 𝑏𝑎𝑏))
87adantl 277 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏𝑎𝑏))
9 onsuc 4501 . . . . 5 (𝑏 ∈ On → suc 𝑏 ∈ On)
10 ordsucunielexmid.1 . . . . . 6 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)
11 eleq1 2240 . . . . . . . 8 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
12 suceq 4403 . . . . . . . . 9 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
1312eleq1d 2246 . . . . . . . 8 (𝑥 = 𝑎 → (suc 𝑥𝑦 ↔ suc 𝑎𝑦))
1411, 13imbi12d 234 . . . . . . 7 (𝑥 = 𝑎 → ((𝑥 𝑦 → suc 𝑥𝑦) ↔ (𝑎 𝑦 → suc 𝑎𝑦)))
15 unieq 3819 . . . . . . . . 9 (𝑦 = suc 𝑏 𝑦 = suc 𝑏)
1615eleq2d 2247 . . . . . . . 8 (𝑦 = suc 𝑏 → (𝑎 𝑦𝑎 suc 𝑏))
17 eleq2 2241 . . . . . . . 8 (𝑦 = suc 𝑏 → (suc 𝑎𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
1816, 17imbi12d 234 . . . . . . 7 (𝑦 = suc 𝑏 → ((𝑎 𝑦 → suc 𝑎𝑦) ↔ (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
1914, 18rspc2va 2856 . . . . . 6 (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
2010, 19mpan2 425 . . . . 5 ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
219, 20sylan2 286 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
228, 21sylbird 170 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏))
2322rgen2a 2531 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ∈ suc 𝑏)
2423onsucelsucexmid 4530 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  wral 2455   cuni 3810  Tr wtr 4102  Ord word 4363  Oncon0 4364  suc csuc 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372
This theorem is referenced by: (None)
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