Definition df-exmid 4313

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 4311 with exmidundif 4324. 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 4314 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 4233, in which case EXMID means that all propositions are decidable (see exmidexmid 4314 and notice that it relies on ax-sep 4233). If we instead work with ax-bdsep 16793, 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 4312	. 2 wff EXMID
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1397	. . . . 5 class 𝑥
4		c0 3512	. . . . . 6 class ∅
5	4	csn 3694	. . . . 5 class {∅}
6	3, 5	wss 3214	. . . 4 wff 𝑥 ⊆ {∅}
7	4, 3	wcel 2205	. . . . 5 wff ∅ ∈ 𝑥
8	7	wdc 842	. . . 4 wff DECID ∅ ∈ 𝑥
9	6, 8	wi 4	. . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
10	9, 2	wal 1396	. 2 wff ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
11	1, 10	wb 105	1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

This definition is referenced by:  exmidexmid  4314  exmid01  4316  exmidsssnc  4321  exmid0el  4322  exmidundif  4324  exmidundifim  4325  exmid1stab  4326  pw1dc0el  7184  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519