Theorem exmidn0m 4333

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 746	. . . 4 ⊢ ((EXMID ∧ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
3		notm0 3542	. . . . . . 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 745	. . . 4 ⊢ ((EXMID ∧ ¬ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
7		exmidexmid 4328	. . . . 5 ⊢ (EXMID → DECID ∃𝑦 𝑦 ∈ 𝑥)
8		exmiddc 848	. . . . 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 810	. . 3 ⊢ (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
11	10	alrimiv 1927	. 2 ⊢ (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
12		orc 724	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
13	12	a1d 22	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		sssnm 3874	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
15	14	biimpa 296	. . . . . . 7 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → 𝑥 = {∅})
16	15	olcd 746	. . . . . 6 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
17	16	ex 115	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
18	13, 17	jaoi 728	. . . 4 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
19	18	alimi 1508	. . 3 ⊢ (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20		exmid01 4330	. . 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 720  DECID wdc 846  ∀wal 1400   = wceq 1402  ∃wex 1545   ⊆ wss 3220  ∅c0 3520  {csn 3705  EXMIDwem 4326

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-exmid 4327

This theorem is referenced by:  exmidsssn  4334