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Theorem onsucsssucexmid 4410
Description: The converse of onsucsssucr 4393 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Hypothesis
Ref Expression
onsucsssucexmid.1  |-  A. x  e.  On  A. y  e.  On  ( x  C_  y  ->  suc  x  C_  suc  y )
Assertion
Ref Expression
onsucsssucexmid  |-  ( ph  \/  -.  ph )
Distinct variable groups:    ph, x    x, y
Allowed substitution hint:    ph( y)

Proof of Theorem onsucsssucexmid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3150 . . . . . 6  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
2 ordtriexmidlem 4403 . . . . . . 7  |-  { z  e.  { (/) }  |  ph }  e.  On
3 sseq1 3088 . . . . . . . . 9  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  C_  {
(/) }  <->  { z  e.  { (/)
}  |  ph }  C_ 
{ (/) } ) )
4 suceq 4292 . . . . . . . . . 10  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  suc  x  =  suc  { z  e.  { (/)
}  |  ph }
)
54sseq1d 3094 . . . . . . . . 9  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( suc  x  C_ 
suc  { (/) }  <->  suc  { z  e.  { (/) }  |  ph }  C_  suc  { (/) } ) )
63, 5imbi12d 233 . . . . . . . 8  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x 
C_  { (/) }  ->  suc  x  C_  suc  { (/) } )  <->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  ->  suc 
{ z  e.  { (/)
}  |  ph }  C_ 
suc  { (/) } ) ) )
7 suc0 4301 . . . . . . . . . 10  |-  suc  (/)  =  { (/)
}
8 0elon 4282 . . . . . . . . . . 11  |-  (/)  e.  On
98onsuci 4400 . . . . . . . . . 10  |-  suc  (/)  e.  On
107, 9eqeltrri 2189 . . . . . . . . 9  |-  { (/) }  e.  On
11 p0ex 4080 . . . . . . . . . 10  |-  { (/) }  e.  _V
12 eleq1 2178 . . . . . . . . . . . 12  |-  ( y  =  { (/) }  ->  ( y  e.  On  <->  { (/) }  e.  On ) )
1312anbi2d 457 . . . . . . . . . . 11  |-  ( y  =  { (/) }  ->  ( ( x  e.  On  /\  y  e.  On )  <-> 
( x  e.  On  /\ 
{ (/) }  e.  On ) ) )
14 sseq2 3089 . . . . . . . . . . . 12  |-  ( y  =  { (/) }  ->  ( x  C_  y  <->  x  C_  { (/) } ) )
15 suceq 4292 . . . . . . . . . . . . 13  |-  ( y  =  { (/) }  ->  suc  y  =  suc  { (/)
} )
1615sseq2d 3095 . . . . . . . . . . . 12  |-  ( y  =  { (/) }  ->  ( suc  x  C_  suc  y 
<->  suc  x  C_  suc  {
(/) } ) )
1714, 16imbi12d 233 . . . . . . . . . . 11  |-  ( y  =  { (/) }  ->  ( ( x  C_  y  ->  suc  x  C_  suc  y )  <->  ( x  C_ 
{ (/) }  ->  suc  x  C_  suc  { (/) } ) ) )
1813, 17imbi12d 233 . . . . . . . . . 10  |-  ( y  =  { (/) }  ->  ( ( ( x  e.  On  /\  y  e.  On )  ->  (
x  C_  y  ->  suc  x  C_  suc  y ) )  <->  ( ( x  e.  On  /\  { (/)
}  e.  On )  ->  ( x  C_  {
(/) }  ->  suc  x  C_ 
suc  { (/) } ) ) ) )
19 onsucsssucexmid.1 . . . . . . . . . . 11  |-  A. x  e.  On  A. y  e.  On  ( x  C_  y  ->  suc  x  C_  suc  y )
2019rspec2 2496 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  C_  y  ->  suc  x  C_  suc  y ) )
2111, 18, 20vtocl 2712 . . . . . . . . 9  |-  ( ( x  e.  On  /\  {
(/) }  e.  On )  ->  ( x  C_  {
(/) }  ->  suc  x  C_ 
suc  { (/) } ) )
2210, 21mpan2 419 . . . . . . . 8  |-  ( x  e.  On  ->  (
x  C_  { (/) }  ->  suc  x  C_  suc  { (/) } ) )
236, 22vtoclga 2724 . . . . . . 7  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  ->  suc 
{ z  e.  { (/)
}  |  ph }  C_ 
suc  { (/) } ) )
242, 23ax-mp 5 . . . . . 6  |-  ( { z  e.  { (/) }  |  ph }  C_  {
(/) }  ->  suc  {
z  e.  { (/) }  |  ph }  C_  suc  { (/) } )
251, 24ax-mp 5 . . . . 5  |-  suc  {
z  e.  { (/) }  |  ph }  C_  suc  { (/) }
2610onsuci 4400 . . . . . . 7  |-  suc  { (/)
}  e.  On
2726onordi 4316 . . . . . 6  |-  Ord  suc  {
(/) }
28 ordelsuc 4389 . . . . . 6  |-  ( ( { z  e.  { (/)
}  |  ph }  e.  On  /\  Ord  suc  {
(/) } )  ->  ( { z  e.  { (/)
}  |  ph }  e.  suc  { (/) }  <->  suc  { z  e.  { (/) }  |  ph }  C_  suc  { (/) } ) )
292, 27, 28mp2an 420 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  suc  { (/) }  <->  suc  { z  e.  { (/) }  |  ph }  C_  suc  { (/) } )
3025, 29mpbir 145 . . . 4  |-  { z  e.  { (/) }  |  ph }  e.  suc  { (/)
}
31 elsucg 4294 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  On  ->  ( { z  e.  { (/) }  |  ph }  e.  suc  { (/)
}  <->  ( { z  e.  { (/) }  |  ph }  e.  { (/) }  \/  { z  e. 
{ (/) }  |  ph }  =  { (/) } ) ) )
322, 31ax-mp 5 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  e.  suc  { (/) }  <->  ( {
z  e.  { (/) }  |  ph }  e.  {
(/) }  \/  { z  e.  { (/) }  |  ph }  =  { (/) } ) )
3330, 32mpbi 144 . . 3  |-  ( { z  e.  { (/) }  |  ph }  e.  {
(/) }  \/  { z  e.  { (/) }  |  ph }  =  { (/) } )
34 elsni 3513 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  e.  {
(/) }  ->  { z  e.  { (/) }  |  ph }  =  (/) )
35 ordtriexmidlem2 4404 . . . . 5  |-  ( { z  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
3634, 35syl 14 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  e.  {
(/) }  ->  -.  ph )
37 0ex 4023 . . . . 5  |-  (/)  e.  _V
38 biidd 171 . . . . 5  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
3937, 38rabsnt 3566 . . . 4  |-  ( { z  e.  { (/) }  |  ph }  =  { (/) }  ->  ph )
4036, 39orim12i 731 . . 3  |-  ( ( { z  e.  { (/)
}  |  ph }  e.  { (/) }  \/  {
z  e.  { (/) }  |  ph }  =  { (/) } )  -> 
( -.  ph  \/  ph ) )
4133, 40ax-mp 5 . 2  |-  ( -. 
ph  \/  ph )
42 orcom 700 . 2  |-  ( ( -.  ph  \/  ph )  <->  (
ph  \/  -.  ph )
)
4341, 42mpbi 144 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   A.wral 2391   {crab 2395    C_ wss 3039   (/)c0 3331   {csn 3495   Ord word 4252   Oncon0 4253   suc csuc 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261
This theorem is referenced by:  oawordriexmid  6332
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