Theorem exmid01 4159

Description: Excluded middle is equivalent to saying any subset of {∅} is either ∅ or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmid01	⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Proof of Theorem exmid01

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-exmid 4156	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2		df-dc 821	. . . . 5 ⊢ (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3		orcom 718	. . . . . 6 ⊢ ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4		simpll 519	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ {∅})
5		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
6	4, 5	sseldd 3129	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ {∅})
7		velsn 3577	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
8	6, 7	sylib 121	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
9	8, 5	eqeltrrd 2235	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥)
10		simplr 520	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ¬ ∅ ∈ 𝑥)
11	9, 10	pm2.65da 651	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦 ∈ 𝑥)
12	11	eq0rdv 3438	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
13	12	ex 114	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
14		noel 3398	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ∅
15		eleq2 2221	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
16	14, 15	mtbiri 665	. . . . . . . 8 ⊢ (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
17	13, 16	impbid1 141	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 ↔ 𝑥 = ∅))
18		ss1o0el1 4158	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (∅ ∈ 𝑥 ↔ 𝑥 = {∅}))
19	17, 18	orbi12d 783	. . . . . 6 ⊢ (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
20	3, 19	syl5bb 191	. . . . 5 ⊢ (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
21	2, 20	syl5bb 191	. . . 4 ⊢ (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
22	21	pm5.74i 179	. . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
23	22	albii 1450	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
24	1, 23	bitri 183	1 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820  ∀wal 1333   = wceq 1335   ∈ wcel 2128   ⊆ wss 3102  ∅c0 3394  {csn 3560  EXMIDwem 4155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-exmid 4156

This theorem is referenced by:  exmid1dc  4161  exmidn0m  4162  exmidsssn  4163  exmidpw  6850  exmidpweq  6851  exmidomni  7080  ss1oel2o  13552  exmidsbthrlem  13580  sbthom  13584