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Theorem ordsucunielexmid 4337
Description: The converse of sucunielr 4317 (where  B is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
Hypothesis
Ref Expression
ordsucunielexmid.1  |-  A. x  e.  On  A. y  e.  On  ( x  e. 
U. y  ->  suc  x  e.  y )
Assertion
Ref Expression
ordsucunielexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x, y

Proof of Theorem ordsucunielexmid
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eloni 4193 . . . . . . . 8  |-  ( b  e.  On  ->  Ord  b )
2 ordtr 4196 . . . . . . . 8  |-  ( Ord  b  ->  Tr  b
)
31, 2syl 14 . . . . . . 7  |-  ( b  e.  On  ->  Tr  b )
4 vex 2622 . . . . . . . 8  |-  b  e. 
_V
54unisuc 4231 . . . . . . 7  |-  ( Tr  b  <->  U. suc  b  =  b )
63, 5sylib 120 . . . . . 6  |-  ( b  e.  On  ->  U. suc  b  =  b )
76eleq2d 2157 . . . . 5  |-  ( b  e.  On  ->  (
a  e.  U. suc  b 
<->  a  e.  b ) )
87adantl 271 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  U. suc  b  <->  a  e.  b ) )
9 suceloni 4308 . . . . 5  |-  ( b  e.  On  ->  suc  b  e.  On )
10 ordsucunielexmid.1 . . . . . 6  |-  A. x  e.  On  A. y  e.  On  ( x  e. 
U. y  ->  suc  x  e.  y )
11 eleq1 2150 . . . . . . . 8  |-  ( x  =  a  ->  (
x  e.  U. y  <->  a  e.  U. y ) )
12 suceq 4220 . . . . . . . . 9  |-  ( x  =  a  ->  suc  x  =  suc  a )
1312eleq1d 2156 . . . . . . . 8  |-  ( x  =  a  ->  ( suc  x  e.  y  <->  suc  a  e.  y ) )
1411, 13imbi12d 232 . . . . . . 7  |-  ( x  =  a  ->  (
( x  e.  U. y  ->  suc  x  e.  y )  <->  ( a  e.  U. y  ->  suc  a  e.  y )
) )
15 unieq 3657 . . . . . . . . 9  |-  ( y  =  suc  b  ->  U. y  =  U. suc  b )
1615eleq2d 2157 . . . . . . . 8  |-  ( y  =  suc  b  -> 
( a  e.  U. y 
<->  a  e.  U. suc  b ) )
17 eleq2 2151 . . . . . . . 8  |-  ( y  =  suc  b  -> 
( suc  a  e.  y 
<->  suc  a  e.  suc  b ) )
1816, 17imbi12d 232 . . . . . . 7  |-  ( y  =  suc  b  -> 
( ( a  e. 
U. y  ->  suc  a  e.  y )  <->  ( a  e.  U. suc  b  ->  suc  a  e.  suc  b ) ) )
1914, 18rspc2va 2734 . . . . . 6  |-  ( ( ( a  e.  On  /\ 
suc  b  e.  On )  /\  A. x  e.  On  A. y  e.  On  ( x  e. 
U. y  ->  suc  x  e.  y )
)  ->  ( a  e.  U. suc  b  ->  suc  a  e.  suc  b ) )
2010, 19mpan2 416 . . . . 5  |-  ( ( a  e.  On  /\  suc  b  e.  On )  ->  ( a  e. 
U. suc  b  ->  suc  a  e.  suc  b
) )
219, 20sylan2 280 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  U. suc  b  ->  suc  a  e.  suc  b ) )
228, 21sylbird 168 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  b  ->  suc  a  e.  suc  b ) )
2322rgen2a 2429 . 2  |-  A. a  e.  On  A. b  e.  On  ( a  e.  b  ->  suc  a  e. 
suc  b )
2423onsucelsucexmid 4336 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   A.wral 2359   U.cuni 3648   Tr wtr 3928   Ord word 4180   Oncon0 4181   suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by: (None)
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