Definition df-exmid 4225

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 4223 with exmidundif 4236. 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 4226 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 4148, in which case EXMID means that all propositions are decidable (see exmidexmid 4226 and notice that it relies on ax-sep 4148). If we instead work with ax-bdsep 15446, 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

Step	Hyp	Ref	Expression
1		wem 4224	. 2 wff EXMID
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1363	. . . . 5 class 𝑥
4		c0 3447	. . . . . 6 class ∅
5	4	csn 3619	. . . . 5 class {∅}
6	3, 5	wss 3154	. . . 4 wff 𝑥 ⊆ {∅}
7	4, 3	wcel 2164	. . . . 5 wff ∅ ∈ 𝑥
8	7	wdc 835	. . . 4 wff DECID ∅ ∈ 𝑥
9	6, 8	wi 4	. . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
10	9, 2	wal 1362	. 2 wff ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
11	1, 10	wb 105	1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

This definition is referenced by:  exmidexmid  4226  exmid01  4228  exmidsssnc  4233  exmid0el  4234  exmidundif  4236  exmidundifim  4237  exmid1stab  4238  pw1dc0el  6969  exmidfodomrlemr  7264  exmidfodomrlemrALT  7265