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Theorem ordsucunielexmid 4284
Description: The converse of sucunielr 4264 (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 4140 . . . . . . . 8  |-  ( b  e.  On  ->  Ord  b )
2 ordtr 4143 . . . . . . . 8  |-  ( Ord  b  ->  Tr  b
)
31, 2syl 14 . . . . . . 7  |-  ( b  e.  On  ->  Tr  b )
4 vex 2577 . . . . . . . 8  |-  b  e. 
_V
54unisuc 4178 . . . . . . 7  |-  ( Tr  b  <->  U. suc  b  =  b )
63, 5sylib 131 . . . . . 6  |-  ( b  e.  On  ->  U. suc  b  =  b )
76eleq2d 2123 . . . . 5  |-  ( b  e.  On  ->  (
a  e.  U. suc  b 
<->  a  e.  b ) )
87adantl 266 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  U. suc  b  <->  a  e.  b ) )
9 suceloni 4255 . . . . 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 2116 . . . . . . . 8  |-  ( x  =  a  ->  (
x  e.  U. y  <->  a  e.  U. y ) )
12 suceq 4167 . . . . . . . . 9  |-  ( x  =  a  ->  suc  x  =  suc  a )
1312eleq1d 2122 . . . . . . . 8  |-  ( x  =  a  ->  ( suc  x  e.  y  <->  suc  a  e.  y ) )
1411, 13imbi12d 227 . . . . . . 7  |-  ( x  =  a  ->  (
( x  e.  U. y  ->  suc  x  e.  y )  <->  ( a  e.  U. y  ->  suc  a  e.  y )
) )
15 unieq 3617 . . . . . . . . 9  |-  ( y  =  suc  b  ->  U. y  =  U. suc  b )
1615eleq2d 2123 . . . . . . . 8  |-  ( y  =  suc  b  -> 
( a  e.  U. y 
<->  a  e.  U. suc  b ) )
17 eleq2 2117 . . . . . . . 8  |-  ( y  =  suc  b  -> 
( suc  a  e.  y 
<->  suc  a  e.  suc  b ) )
1816, 17imbi12d 227 . . . . . . 7  |-  ( y  =  suc  b  -> 
( ( a  e. 
U. y  ->  suc  a  e.  y )  <->  ( a  e.  U. suc  b  ->  suc  a  e.  suc  b ) ) )
1914, 18rspc2va 2686 . . . . . 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 409 . . . . 5  |-  ( ( a  e.  On  /\  suc  b  e.  On )  ->  ( a  e. 
U. suc  b  ->  suc  a  e.  suc  b
) )
219, 20sylan2 274 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  U. suc  b  ->  suc  a  e.  suc  b ) )
228, 21sylbird 163 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( a  e.  b  ->  suc  a  e.  suc  b ) )
2322rgen2a 2392 . 2  |-  A. a  e.  On  A. b  e.  On  ( a  e.  b  ->  suc  a  e. 
suc  b )
2423onsucelsucexmid 4283 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   A.wral 2323   U.cuni 3608   Tr wtr 3882   Ord word 4127   Oncon0 4128   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by: (None)
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