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

Theorem exmidexmid 4182
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 838, peircedc 909, or condc 848.

(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 3232 . . 3  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
2 df-exmid 4181 . . . 4  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
3 p0ex 4174 . . . . . 6  |-  { (/) }  e.  _V
43rabex 4133 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  _V
5 sseq1 3170 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  C_  {
(/) }  <->  { z  e.  { (/)
}  |  ph }  C_ 
{ (/) } ) )
6 eleq2 2234 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
76dcbid 833 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  (DECID  (/)  e.  x  <-> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
85, 7imbi12d 233 . . . . 5  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x 
C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) ) )
94, 8spcv 2824 . . . 4  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  ->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
102, 9sylbi 120 . . 3  |-  (EXMID  ->  ( { z  e.  { (/)
}  |  ph }  C_ 
{ (/) }  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
111, 10mpi 15 . 2  |-  (EXMID  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } )
12 0ex 4116 . . . . 5  |-  (/)  e.  _V
1312snid 3614 . . . 4  |-  (/)  e.  { (/)
}
14 biidd 171 . . . . 5  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1514elrab 2886 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  (
(/)  e.  { (/) }  /\  ph ) )
1613, 15mpbiran 935 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
1716dcbii 835 . 2  |-  (DECID  (/)  e.  {
z  e.  { (/) }  |  ph }  <-> DECID  ph )
1811, 17sylib 121 1  |-  (EXMID  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 829   A.wal 1346    = wceq 1348    e. wcel 2141   {crab 2452    C_ wss 3121   (/)c0 3414   {csn 3583  EXMIDwem 4180
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-exmid 4181
This theorem is referenced by:  exmidn0m  4187  exmid0el  4190  exmidel  4191  exmidundif  4192  exmidundifim  4193  sbthlemi3  6936  sbthlemi5  6938  sbthlemi6  6939  exmidomniim  7117  exmidfodomrlemim  7178  exmidontriimlem1  7198
  Copyright terms: Public domain W3C validator