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Theorem ordsucunielexmid 4216
Description: The converse of sucunielr 4201 (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 x y)
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 4078 . . . . . . . 8 (𝑏 On → Ord 𝑏)
2 ordtr 4081 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 On → Tr 𝑏)
4 vex 2554 . . . . . . . 8 𝑏 V
54unisuc 4116 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 127 . . . . . 6 (𝑏 On → suc 𝑏 = 𝑏)
76eleq2d 2104 . . . . 5 (𝑏 On → (𝑎 suc 𝑏𝑎 𝑏))
87adantl 262 . . . 4 ((𝑎 On 𝑏 On) → (𝑎 suc 𝑏𝑎 𝑏))
9 suceloni 4193 . . . . 5 (𝑏 On → suc 𝑏 On)
10 ordsucunielexmid.1 . . . . . 6 x On y On (x y → suc x y)
11 eleq1 2097 . . . . . . . 8 (x = 𝑎 → (x y𝑎 y))
12 suceq 4105 . . . . . . . . 9 (x = 𝑎 → suc x = suc 𝑎)
1312eleq1d 2103 . . . . . . . 8 (x = 𝑎 → (suc x y ↔ suc 𝑎 y))
1411, 13imbi12d 223 . . . . . . 7 (x = 𝑎 → ((x y → suc x y) ↔ (𝑎 y → suc 𝑎 y)))
15 unieq 3580 . . . . . . . . 9 (y = suc 𝑏 y = suc 𝑏)
1615eleq2d 2104 . . . . . . . 8 (y = suc 𝑏 → (𝑎 y𝑎 suc 𝑏))
17 eleq2 2098 . . . . . . . 8 (y = suc 𝑏 → (suc 𝑎 y ↔ suc 𝑎 suc 𝑏))
1816, 17imbi12d 223 . . . . . . 7 (y = suc 𝑏 → ((𝑎 y → suc 𝑎 y) ↔ (𝑎 suc 𝑏 → suc 𝑎 suc 𝑏)))
1914, 18rspc2va 2657 . . . . . 6 (((𝑎 On suc 𝑏 On) x On y On (x y → suc x y)) → (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) → (𝑎 𝑏 → suc 𝑎 suc 𝑏))
2322rgen2a 2369 . 2 𝑎 On 𝑏 On (𝑎 𝑏 → suc 𝑎 suc 𝑏)
2423onsucelsucexmid 4215 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  wral 2300   cuni 3571  Tr wtr 3845  Ord word 4065  Oncon0 4066  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
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