Theorem exmidexmid 4288

Description: EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 850, peircedc 921, or condc 860.

(Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmidexmid	⊢ (EXMID → DECID 𝜑)

Proof of Theorem exmidexmid

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3311	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		df-exmid 4287	. . . 4 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
3		p0ex 4280	. . . . . 6 ⊢ {∅} ∈ V
4	3	rabex 4235	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
5		sseq1 3249	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
6		eleq2 2294	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
7	6	dcbid 845	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (DECID ∅ ∈ 𝑥 ↔ DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
8	5, 7	imbi12d 234	. . . . 5 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9	4, 8	spcv 2899	. . . 4 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
10	2, 9	sylbi 121	. . 3 ⊢ (EXMID → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
11	1, 10	mpi 15	. 2 ⊢ (EXMID → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
12		0ex 4217	. . . . 5 ⊢ ∅ ∈ V
13	12	snid 3701	. . . 4 ⊢ ∅ ∈ {∅}
14		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
15	14	elrab 2961	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
16	13, 15	mpbiran 948	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
17	16	dcbii 847	. 2 ⊢ (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑)
18	11, 17	sylib 122	1 ⊢ (EXMID → DECID 𝜑)

Syntax hints:   → wi 4  DECID wdc 841  ∀wal 1395   = wceq 1397   ∈ wcel 2201  {crab 2513   ⊆ wss 3199  ∅c0 3493  {csn 3670  EXMIDwem 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rab 2518  df-v 2803  df-dif 3201  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-exmid 4287

This theorem is referenced by:  exmidn0m  4293  exmid0el  4296  exmidel  4297  exmidundif  4298  exmidundifim  4299  exmidpw2en  7109  exmidssfi  7136  sbthlemi3  7163  sbthlemi5  7165  sbthlemi6  7166  exmidomniim  7345  exmidfodomrlemim  7417  exmidontriimlem1  7441  exmidapne  7484  pw1dceq  16665  exmidnotnotr  16666  exmidcon  16667  exmidpeirce  16668