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Definition df-exmid 4114
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 4112 with exmidundif 4124. 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 𝜑 and ¬ 𝜑 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 𝜑 ∨ ¬ 𝜑 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 𝜑 by exmidexmid 4115 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 4041, in which case EXMID means that all propositions are decidable (see exmidexmid 4115 and notice that it relies on ax-sep 4041). If we instead work with ax-bdsep 13071, 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 ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

Detailed syntax breakdown of Definition df-exmid
StepHypRef Expression
1 wem 4113 . 2 wff EXMID
2 vx . . . . . 6 setvar 𝑥
32cv 1330 . . . . 5 class 𝑥
4 c0 3358 . . . . . 6 class
54csn 3522 . . . . 5 class {∅}
63, 5wss 3066 . . . 4 wff 𝑥 ⊆ {∅}
74, 3wcel 1480 . . . . 5 wff ∅ ∈ 𝑥
87wdc 819 . . . 4 wff DECID ∅ ∈ 𝑥
96, 8wi 4 . . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
109, 2wal 1329 . 2 wff 𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
111, 10wb 104 1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
Colors of variables: wff set class
This definition is referenced by:  exmidexmid  4115  exmid01  4116  exmidsssnc  4121  exmid0el  4122  exmidundif  4124  exmidundifim  4125  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  exmid1stab  13184
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