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Theorem ordsucunielexmid 4200
Description: The converse of sucunielr 4185 (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 4061 . . . . . . . 8  b  On  Ord  b
2 ordtr 4064 . . . . . . . 8  Ord  b  Tr  b
31, 2syl 14 . . . . . . 7  b  On  Tr  b
4 vex 2538 . . . . . . . 8  b 
_V
54unisuc 4099 . . . . . . 7  Tr  b  U. suc  b  b
63, 5sylib 127 . . . . . 6  b  On  U. suc  b  b
76eleq2d 2089 . . . . 5  b  On 
a  U. suc  b  a  b
87adantl 262 . . . 4  a  On  b  On  a  U. suc  b  a  b
9 suceloni 4177 . . . . 5  b  On  suc  b  On
10 ordsucunielexmid.1 . . . . . 6  On  On 
U.  suc
11 eleq1 2082 . . . . . . . 8  a  U.  a  U.
12 suceq 4088 . . . . . . . . 9  a  suc  suc  a
1312eleq1d 2088 . . . . . . . 8  a  suc  suc  a
1411, 13imbi12d 223 . . . . . . 7  a  U.  suc  a 
U.  suc  a
15 unieq 3563 . . . . . . . . 9  suc  b  U.  U. suc  b
1615eleq2d 2089 . . . . . . . 8  suc  b  a  U.  a  U. suc  b
17 eleq2 2083 . . . . . . . 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 2640 . . . . . 6  a  On 
suc  b  On  On  On 
U.  suc  a 
U. suc  b  suc  a  suc  b
2010, 19mpan2 403 . . . . 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 2353 . 2  a  On  b  On  a  b  suc  a  suc  b
2423onsucelsucexmid 4199 1
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 616   wceq 1228   wcel 1374  wral 2284   U.cuni 3554   Tr wtr 3828   Ord word 4048   Oncon0 4049   suc csuc 4051
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by: (None)
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