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Definition df-exmid 4192
Description: The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions.

To see how this compares with other ways of expressing excluded middle, compare undifexmid 4190 with exmidundif 4203. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show  ph and  -.  ph in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis  ph  \/  -.  ph for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID  ph by exmidexmid 4193 but there is no good way to express the converse.

This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4118, in which case EXMID means that all propositions are decidable (see exmidexmid 4193 and notice that it relies on ax-sep 4118). If we instead work with ax-bdsep 14285, EXMID as defined here means that all bounded propositions are decidable.

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

Assertion
Ref Expression
df-exmid  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )

Detailed syntax breakdown of Definition df-exmid
StepHypRef Expression
1 wem 4191 . 2  wff EXMID
2 vx . . . . . 6  setvar  x
32cv 1352 . . . . 5  class  x
4 c0 3422 . . . . . 6  class  (/)
54csn 3591 . . . . 5  class  { (/) }
63, 5wss 3129 . . . 4  wff  x  C_  {
(/) }
74, 3wcel 2148 . . . . 5  wff  (/)  e.  x
87wdc 834 . . . 4  wff DECID  (/)  e.  x
96, 8wi 4 . . 3  wff  ( x 
C_  { (/) }  -> DECID  (/)  e.  x
)
109, 2wal 1351 . 2  wff  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x )
111, 10wb 105 1  wff  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
Colors of variables: wff set class
This definition is referenced by:  exmidexmid  4193  exmid01  4195  exmidsssnc  4200  exmid0el  4201  exmidundif  4203  exmidundifim  4204  exmid1stab  4205  pw1dc0el  6905  exmidfodomrlemr  7195  exmidfodomrlemrALT  7196
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