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Theorem ordsucunielexmid 4216
Description: The converse of sucunielr 4201 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
Hypothesis
Ref Expression
ordsucunielexmid.1  On  On 
U.  suc
Assertion
Ref Expression
ordsucunielexmid
Distinct variable group:   ,,

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  U. suc  b  b
63, 5sylib 127 . . . . . 6  b  On  U. suc  b  b
76eleq2d 2104 . . . . 5  b  On 
a  U. suc  b  a  b
87adantl 262 . . . 4  a  On  b  On  a  U. suc  b  a  b
9 suceloni 4193 . . . . 5  b  On  suc  b  On
10 ordsucunielexmid.1 . . . . . 6  On  On 
U.  suc
11 eleq1 2097 . . . . . . . 8  a  U.  a  U.
12 suceq 4105 . . . . . . . . 9  a  suc  suc  a
1312eleq1d 2103 . . . . . . . 8  a  suc  suc  a
1411, 13imbi12d 223 . . . . . . 7  a  U.  suc  a 
U.  suc  a
15 unieq 3580 . . . . . . . . 9  suc  b  U.  U. suc  b
1615eleq2d 2104 . . . . . . . 8  suc  b  a  U.  a  U. suc  b
17 eleq2 2098 . . . . . . . 8  suc  b  suc  a  suc  a  suc  b
1816, 17imbi12d 223 . . . . . . 7  suc  b  a  U.  suc  a  a  U. suc  b  suc  a  suc  b
1914, 18rspc2va 2657 . . . . . 6  a  On 
suc  b  On  On  On 
U.  suc  a 
U. suc  b  suc  a  suc  b
2010, 19mpan2 401 . . . . 5  a  On  suc  b  On  a 
U. suc  b  suc  a  suc  b
219, 20sylan2 270 . . . 4  a  On  b  On  a  U. 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
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390  wral 2300   U.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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