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Theorem ordsucunielexmid 4173
Description: The converse of sucunielr 4158 (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 4035 . . . . . . . 8 (𝑏 On → Ord 𝑏)
2 ordtr 4038 . . . . . . . 8 (Ord 𝑏 → Tr 𝑏)
31, 2syl 14 . . . . . . 7 (𝑏 On → Tr 𝑏)
4 vex 2536 . . . . . . . 8 𝑏 V
54unisuc 4073 . . . . . . 7 (Tr 𝑏 suc 𝑏 = 𝑏)
63, 5sylib 127 . . . . . 6 (𝑏 On → suc 𝑏 = 𝑏)
76eleq2d 2089 . . . . 5 (𝑏 On → (𝑎 suc 𝑏𝑎 𝑏))
87adantl 262 . . . 4 ((𝑎 On 𝑏 On) → (𝑎 suc 𝑏𝑎 𝑏))
9 suceloni 4150 . . . . 5 (𝑏 On → suc 𝑏 On)
10 ordsucunielexmid.1 . . . . . 6 x On y On (x y → suc x y)
11 eleq1 2082 . . . . . . . 8 (x = 𝑎 → (x y𝑎 y))
12 suceq 4062 . . . . . . . . 9 (x = 𝑎 → suc x = suc 𝑎)
1312eleq1d 2088 . . . . . . . 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 3541 . . . . . . . . 9 (y = suc 𝑏 y = suc 𝑏)
1615eleq2d 2089 . . . . . . . 8 (y = suc 𝑏 → (𝑎 y𝑎 suc 𝑏))
17 eleq2 2083 . . . . . . . 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 2638 . . . . . 6 (((𝑎 On suc 𝑏 On) x On y On (x y → suc x y)) → (𝑎 suc 𝑏 → suc 𝑎 suc 𝑏))
2010, 19mpan2 403 . . . . 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 2351 . 2 𝑎 On 𝑏 On (𝑎 𝑏 → suc 𝑎 suc 𝑏)
2423onsucelsucexmid 4172 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   = wceq 1373   wcel 1375  wral 2282   cuni 3532  Tr wtr 3806  Ord word 4023  Oncon0 4024  suc csuc 4026
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-nul 3835  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031
This theorem is referenced by: (None)
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