ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0elsucexmid Unicode version

Theorem 0elsucexmid 4576
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 4530 . . . 4  |-  { y  e.  { (/) }  |  ph }  e.  On
2 0elsucexmid.1 . . . 4  |-  A. x  e.  On  (/)  e.  suc  x
3 suceq 4414 . . . . . 6  |-  ( x  =  { y  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { y  e.  { (/)
}  |  ph }
)
43eleq2d 2257 . . . . 5  |-  ( x  =  { y  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  suc  x 
<->  (/)  e.  suc  { y  e.  { (/) }  |  ph } ) )
54rspcv 2849 . . . 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 4142 . . . 4  |-  (/)  e.  _V
87elsuc 4418 . . 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 3635 . . . . 5  |-  (/)  e.  { (/)
}
11 biidd 172 . . . . . 6  |-  ( y  =  (/)  ->  ( ph  <->  ph ) )
1211elrab3 2906 . . . . 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 4531 . . . 4  |-  ( { y  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
1615eqcoms 2190 . . 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 1363    e. wcel 2158   A.wral 2465   {crab 2469   (/)c0 3434   {csn 3604   Oncon0 4375   suc csuc 4377
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator