Definition df-exmid 4258

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 4256 with exmidundif 4269. 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 4259 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 4181, in which case EXMID means that all propositions are decidable (see exmidexmid 4259 and notice that it relies on ax-sep 4181). If we instead work with ax-bdsep 16157, 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 4257	. 2 wff EXMID
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1374	. . . . 5 class 𝑥
4		c0 3471	. . . . . 6 class ∅
5	4	csn 3646	. . . . 5 class {∅}
6	3, 5	wss 3177	. . . 4 wff 𝑥 ⊆ {∅}
7	4, 3	wcel 2180	. . . . 5 wff ∅ ∈ 𝑥
8	7	wdc 838	. . . 4 wff DECID ∅ ∈ 𝑥
9	6, 8	wi 4	. . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
10	9, 2	wal 1373	. 2 wff ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
11	1, 10	wb 105	1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

This definition is referenced by:  exmidexmid  4259  exmid01  4261  exmidsssnc  4266  exmid0el  4267  exmidundif  4269  exmidundifim  4270  exmid1stab  4271  pw1dc0el  7041  exmidfodomrlemr  7348  exmidfodomrlemrALT  7349