Theorem exmidn0m 4222

Description: Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)

Assertion

Ref	Expression
exmidn0m	⊢ (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidn0m

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦 𝑦 ∈ 𝑥)
2	1	olcd 735	. . . 4 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
3		notm0 3458	. . . . . . 7 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
4	3	biimpi 120	. . . . . 6 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 → 𝑥 = ∅)
5	4	adantl 277	. . . . 5 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → 𝑥 = ∅)
6	5	orcd 734	. . . 4 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
7		exmidexmid 4217	. . . . 5 ⊢ (EXMID → DECID ∃𝑦 𝑦 ∈ 𝑥)
8		exmiddc 837	. . . . 5 ⊢ (DECID ∃𝑦 𝑦 ∈ 𝑥 → (∃𝑦 𝑦 ∈ 𝑥 ∨ ¬ ∃𝑦 𝑦 ∈ 𝑥))
9	7, 8	syl 14	. . . 4 ⊢ (EXMID → (∃𝑦 𝑦 ∈ 𝑥 ∨ ¬ ∃𝑦 𝑦 ∈ 𝑥))
10	2, 6, 9	mpjaodan 799	. . 3 ⊢ (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
11	10	alrimiv 1885	. 2 ⊢ (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
12		orc 713	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
13	12	a1d 22	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		sssnm 3772	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
15	14	biimpa 296	. . . . . . 7 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → 𝑥 = {∅})
16	15	olcd 735	. . . . . 6 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
17	16	ex 115	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
18	13, 17	jaoi 717	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
19	18	alimi 1466	. . 3 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20		exmid01 4219	. . 3 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
21	19, 20	sylibr 134	. 2 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → EXMID)
22	11, 21	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835  ∀wal 1362   = wceq 1364  ∃wex 1503   ⊆ wss 3144  ∅c0 3437  {csn 3610  EXMIDwem 4215

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-exmid 4216

This theorem is referenced by:  exmidsssn  4223