Theorem exmidn0m 4314

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 742	. . . 4 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
3		notm0 3529	. . . . . . 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 741	. . . 4 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
7		exmidexmid 4309	. . . . 5 ⊢ (EXMID → DECID ∃𝑦 𝑦 ∈ 𝑥)
8		exmiddc 844	. . . . 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 806	. . 3 ⊢ (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
11	10	alrimiv 1923	. 2 ⊢ (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
12		orc 720	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
13	12	a1d 22	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		sssnm 3858	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
15	14	biimpa 296	. . . . . . 7 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → 𝑥 = {∅})
16	15	olcd 742	. . . . . 6 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
17	16	ex 115	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
18	13, 17	jaoi 724	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
19	18	alimi 1504	. . 3 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20		exmid01 4311	. . 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 716  DECID wdc 842  ∀wal 1396   = wceq 1398  ∃wex 1541   ⊆ wss 3211  ∅c0 3508  {csn 3689  EXMIDwem 4307

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-exmid 4308

This theorem is referenced by:  exmidsssn  4315