Theorem exmidexmid 4180

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 838, peircedc 909, or condc 848.

(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 3232	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		df-exmid 4179	. . . 4 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
3		p0ex 4172	. . . . . 6 ⊢ {∅} ∈ V
4	3	rabex 4131	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
5		sseq1 3170	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
6		eleq2 2234	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
7	6	dcbid 833	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (DECID ∅ ∈ 𝑥 ↔ DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
8	5, 7	imbi12d 233	. . . . 5 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9	4, 8	spcv 2824	. . . 4 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
10	2, 9	sylbi 120	. . 3 ⊢ (EXMID → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
11	1, 10	mpi 15	. 2 ⊢ (EXMID → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
12		0ex 4114	. . . . 5 ⊢ ∅ ∈ V
13	12	snid 3612	. . . 4 ⊢ ∅ ∈ {∅}
14		biidd 171	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
15	14	elrab 2886	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
16	13, 15	mpbiran 935	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
17	16	dcbii 835	. 2 ⊢ (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑)
18	11, 17	sylib 121	1 ⊢ (EXMID → DECID 𝜑)

Syntax hints:   → wi 4  DECID wdc 829  ∀wal 1346   = wceq 1348   ∈ wcel 2141  {crab 2452   ⊆ wss 3121  ∅c0 3414  {csn 3581  EXMIDwem 4178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-exmid 4179

This theorem is referenced by:  exmidn0m  4185  exmid0el  4188  exmidel  4189  exmidundif  4190  exmidundifim  4191  sbthlemi3  6932  sbthlemi5  6934  sbthlemi6  6935  exmidomniim  7113  exmidfodomrlemim  7165  exmidontriimlem1  7185