Definition df-exmid 4181

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 4179 with exmidundif 4192. 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 4182 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 4107, in which case EXMID means that all propositions are decidable (see exmidexmid 4182 and notice that it relies on ax-sep 4107). If we instead work with ax-bdsep 13919, 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 4180	. 2 wff EXMID
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1347	. . . . 5 class 𝑥
4		c0 3414	. . . . . 6 class ∅
5	4	csn 3583	. . . . 5 class {∅}
6	3, 5	wss 3121	. . . 4 wff 𝑥 ⊆ {∅}
7	4, 3	wcel 2141	. . . . 5 wff ∅ ∈ 𝑥
8	7	wdc 829	. . . 4 wff DECID ∅ ∈ 𝑥
9	6, 8	wi 4	. . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
10	9, 2	wal 1346	. 2 wff ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
11	1, 10	wb 104	1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

This definition is referenced by:  exmidexmid  4182  exmid01  4184  exmidsssnc  4189  exmid0el  4190  exmidundif  4192  exmidundifim  4193  pw1dc0el  6889  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  exmid1stab  14033