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

Theorem exmidexmid 4193
Description: EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 843, peircedc 914, or condc 853.

(Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion
Ref Expression
exmidexmid  |-  (EXMID  -> DECID  ph )

Proof of Theorem exmidexmid
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3240 . . 3  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
2 df-exmid 4192 . . . 4  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
3 p0ex 4185 . . . . . 6  |-  { (/) }  e.  _V
43rabex 4144 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  _V
5 sseq1 3178 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  C_  {
(/) }  <->  { z  e.  { (/)
}  |  ph }  C_ 
{ (/) } ) )
6 eleq2 2241 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
76dcbid 838 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  (DECID  (/)  e.  x  <-> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
85, 7imbi12d 234 . . . . 5  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x 
C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) ) )
94, 8spcv 2831 . . . 4  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  ->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
102, 9sylbi 121 . . 3  |-  (EXMID  ->  ( { z  e.  { (/)
}  |  ph }  C_ 
{ (/) }  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
111, 10mpi 15 . 2  |-  (EXMID  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } )
12 0ex 4127 . . . . 5  |-  (/)  e.  _V
1312snid 3622 . . . 4  |-  (/)  e.  { (/)
}
14 biidd 172 . . . . 5  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1514elrab 2893 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  (
(/)  e.  { (/) }  /\  ph ) )
1613, 15mpbiran 940 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
1716dcbii 840 . 2  |-  (DECID  (/)  e.  {
z  e.  { (/) }  |  ph }  <-> DECID  ph )
1811, 17sylib 122 1  |-  (EXMID  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 834   A.wal 1351    = wceq 1353    e. wcel 2148   {crab 2459    C_ wss 3129   (/)c0 3422   {csn 3591  EXMIDwem 4191
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-exmid 4192
This theorem is referenced by:  exmidn0m  4198  exmid0el  4201  exmidel  4202  exmidundif  4203  exmidundifim  4204  sbthlemi3  6951  sbthlemi5  6953  sbthlemi6  6954  exmidomniim  7132  exmidfodomrlemim  7193  exmidontriimlem1  7213
  Copyright terms: Public domain W3C validator