Theorem exmidexmid 4259

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 847, peircedc 918, or condc 857.

(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 3289	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		df-exmid 4258	. . . 4 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
3		p0ex 4251	. . . . . 6 ⊢ {∅} ∈ V
4	3	rabex 4207	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
5		sseq1 3227	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
6		eleq2 2273	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
7	6	dcbid 842	. . . . . 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 2877	. . . 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 4190	. . . . 5 ⊢ ∅ ∈ V
13	12	snid 3677	. . . 4 ⊢ ∅ ∈ {∅}
14		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
15	14	elrab 2939	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
16	13, 15	mpbiran 945	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
17	16	dcbii 844	. 2 ⊢ (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑)
18	11, 17	sylib 122	1 ⊢ (EXMID → DECID 𝜑)

Syntax hints:   → wi 4  DECID wdc 838  ∀wal 1373   = wceq 1375   ∈ wcel 2180  {crab 2492   ⊆ wss 3177  ∅c0 3471  {csn 3646  EXMIDwem 4257

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237

This theorem depends on definitions:  df-bi 117  df-dc 839  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rab 2497  df-v 2781  df-dif 3179  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-exmid 4258

This theorem is referenced by:  exmidn0m  4264  exmid0el  4267  exmidel  4268  exmidundif  4269  exmidundifim  4270  exmidpw2en  7042  sbthlemi3  7094  sbthlemi5  7096  sbthlemi6  7097  exmidomniim  7276  exmidfodomrlemim  7347  exmidontriimlem1  7371  exmidapne  7414