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Theorem 0elsucexmid 4579
Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
Hypothesis
Ref Expression
0elsucexmid.1  |-  A. x  e.  On  (/)  e.  suc  x
Assertion
Ref Expression
0elsucexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    ph, x

Proof of Theorem 0elsucexmid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ordtriexmidlem 4533 . . . 4  |-  { y  e.  { (/) }  |  ph }  e.  On
2 0elsucexmid.1 . . . 4  |-  A. x  e.  On  (/)  e.  suc  x
3 suceq 4417 . . . . . 6  |-  ( x  =  { y  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { y  e.  { (/)
}  |  ph }
)
43eleq2d 2259 . . . . 5  |-  ( x  =  { y  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  suc  x 
<->  (/)  e.  suc  { y  e.  { (/) }  |  ph } ) )
54rspcv 2852 . . . 4  |-  ( { y  e.  { (/) }  |  ph }  e.  On  ->  ( A. x  e.  On  (/)  e.  suc  x  -> 
(/)  e.  suc  { y  e.  { (/) }  |  ph } ) )
61, 2, 5mp2 16 . . 3  |-  (/)  e.  suc  { y  e.  { (/) }  |  ph }
7 0ex 4145 . . . 4  |-  (/)  e.  _V
87elsuc 4421 . . 3  |-  ( (/)  e.  suc  { y  e. 
{ (/) }  |  ph } 
<->  ( (/)  e.  { y  e.  { (/) }  |  ph }  \/  (/)  =  {
y  e.  { (/) }  |  ph } ) )
96, 8mpbi 145 . 2  |-  ( (/)  e.  { y  e.  { (/)
}  |  ph }  \/  (/)  =  { y  e.  { (/) }  |  ph } )
107snid 3638 . . . . 5  |-  (/)  e.  { (/)
}
11 biidd 172 . . . . . 6  |-  ( y  =  (/)  ->  ( ph  <->  ph ) )
1211elrab3 2909 . . . . 5  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { y  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
1310, 12ax-mp 5 . . . 4  |-  ( (/)  e.  { y  e.  { (/)
}  |  ph }  <->  ph )
1413biimpi 120 . . 3  |-  ( (/)  e.  { y  e.  { (/)
}  |  ph }  ->  ph )
15 ordtriexmidlem2 4534 . . . 4  |-  ( { y  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
1615eqcoms 2192 . . 3  |-  ( (/)  =  { y  e.  { (/)
}  |  ph }  ->  -.  ph )
1714, 16orim12i 760 . 2  |-  ( (
(/)  e.  { y  e.  { (/) }  |  ph }  \/  (/)  =  {
y  e.  { (/) }  |  ph } )  ->  ( ph  \/  -.  ph ) )
189, 17ax-mp 5 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   A.wral 2468   {crab 2472   (/)c0 3437   {csn 3607   Oncon0 4378   suc csuc 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386
This theorem is referenced by: (None)
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