Theorem exmidexmid 4331

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 855, peircedc 926, or condc 865.

(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 3333	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		df-exmid 4330	. . . 4 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
3		p0ex 4323	. . . . . 6 ⊢ {∅} ∈ V
4	3	rabex 4278	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
5		sseq1 3271	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
6		eleq2 2302	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
7	6	dcbid 850	. . . . . 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 2919	. . . 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 4258	. . . . 5 ⊢ ∅ ∈ V
13	12	snid 3739	. . . 4 ⊢ ∅ ∈ {∅}
14		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
15	14	elrab 2982	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
16	13, 15	mpbiran 953	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
17	16	dcbii 852	. 2 ⊢ (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑)
18	11, 17	sylib 122	1 ⊢ (EXMID → DECID 𝜑)

Syntax hints:   → wi 4  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {crab 2532   ⊆ wss 3220  ∅c0 3520  {csn 3708  EXMIDwem 4329

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-exmid 4330

This theorem is referenced by:  exmidn0m  4336  exmid0el  4339  exmidel  4340  exmidundif  4341  exmidundifim  4342  exmidpw2en  7213  exmidssfi  7240  sbthlemi3  7270  sbthlemi5  7272  sbthlemi6  7273  exmidomniim  7475  exmidfodomrlemim  7547  exmidontriimlem1  7571  exmidapne  7620  pw1dceq  17017  exmidnotnotr  17018  exmidcon  17019  exmidpeirce  17020