Theorem exmidn0m 4054

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 109	. . . . 5 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦 𝑦 ∈ 𝑥)
2	1	olcd 691	. . . 4 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
3		notm0 3322	. . . . . . 7 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
4	3	biimpi 119	. . . . . 6 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 → 𝑥 = ∅)
5	4	adantl 272	. . . . 5 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → 𝑥 = ∅)
6	5	orcd 690	. . . 4 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
7		exmidexmid 4052	. . . . 5 ⊢ (EXMID → DECID ∃𝑦 𝑦 ∈ 𝑥)
8		exmiddc 785	. . . . 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 750	. . 3 ⊢ (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
11	10	alrimiv 1809	. 2 ⊢ (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
12		orc 671	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
13	12	a1d 22	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		sssnm 3620	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
15	14	biimpa 291	. . . . . . 7 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → 𝑥 = {∅})
16	15	olcd 691	. . . . . 6 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
17	16	ex 114	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
18	13, 17	jaoi 674	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
19	18	alimi 1396	. . 3 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20		exmid01 4053	. . 3 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
21	19, 20	sylibr 133	. 2 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → EXMID)
22	11, 21	impbii 125	1 ⊢ (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667  DECID wdc 783  ∀wal 1294   = wceq 1296  ∃wex 1433   ⊆ wss 3013  ∅c0 3302  {csn 3466  EXMIDwem 4050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030

This theorem depends on definitions:  df-bi 116  df-dc 784  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-exmid 4051

This theorem is referenced by:  exmidsssn  4055