Definition df-exmid 4330

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 4328 with exmidundif 4341. 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 4331 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 4247, in which case EXMID means that all propositions are decidable (see exmidexmid 4331 and notice that it relies on ax-sep 4247). If we instead work with ax-bdsep 16893, 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 4329	. 2 wff EXMID
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1401	. . . . 5 class 𝑥
4		c0 3520	. . . . . 6 class ∅
5	4	csn 3708	. . . . 5 class {∅}
6	3, 5	wss 3220	. . . 4 wff 𝑥 ⊆ {∅}
7	4, 3	wcel 2209	. . . . 5 wff ∅ ∈ 𝑥
8	7	wdc 846	. . . 4 wff DECID ∅ ∈ 𝑥
9	6, 8	wi 4	. . 3 wff (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
10	9, 2	wal 1400	. 2 wff ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)
11	1, 10	wb 105	1 wff (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

This definition is referenced by:  exmidexmid  4331  exmid01  4333  exmidsssnc  4338  exmid0el  4339  exmidundif  4341  exmidundifim  4342  exmid1stab  4343  pw1dc0el  7212  exmidfodomrlemr  7548  exmidfodomrlemrALT  7549