Theorem exmid01 4053

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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-exmid 4051	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2		df-dc 784	. . . . 5 ⊢ (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3		orcom 685	. . . . . 6 ⊢ ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4		simpll 497	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ {∅})
5		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
6	4, 5	sseldd 3040	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ {∅})
7		velsn 3483	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
8	6, 7	sylib 121	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
9	8, 5	eqeltrrd 2172	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥)
10		simplr 498	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ¬ ∅ ∈ 𝑥)
11	9, 10	pm2.65da 625	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦 ∈ 𝑥)
12	11	eq0rdv 3346	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
13	12	ex 114	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
14		noel 3306	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ∅
15		eleq2 2158	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
16	14, 15	mtbiri 638	. . . . . . . 8 ⊢ (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
17	13, 16	impbid1 141	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 ↔ 𝑥 = ∅))
18		elex2 2649	. . . . . . . . . 10 ⊢ (∅ ∈ 𝑥 → ∃𝑧 𝑧 ∈ 𝑥)
19		sssnm 3620	. . . . . . . . . 10 ⊢ (∃𝑧 𝑧 ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (∅ ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
21	20	biimpcd 158	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (∅ ∈ 𝑥 → 𝑥 = {∅}))
22		0ex 3987	. . . . . . . . . 10 ⊢ ∅ ∈ V
23	22	snid 3495	. . . . . . . . 9 ⊢ ∅ ∈ {∅}
24		eleq2 2158	. . . . . . . . 9 ⊢ (𝑥 = {∅} → (∅ ∈ 𝑥 ↔ ∅ ∈ {∅}))
25	23, 24	mpbiri 167	. . . . . . . 8 ⊢ (𝑥 = {∅} → ∅ ∈ 𝑥)
26	21, 25	impbid1 141	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (∅ ∈ 𝑥 ↔ 𝑥 = {∅}))
27	17, 26	orbi12d 745	. . . . . 6 ⊢ (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
28	3, 27	syl5bb 191	. . . . 5 ⊢ (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
29	2, 28	syl5bb 191	. . . 4 ⊢ (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
30	29	pm5.74i 179	. . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
31	30	albii 1411	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
32	1, 31	bitri 183	1 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667  DECID wdc 783  ∀wal 1294   = wceq 1296  ∃wex 1433   ∈ wcel 1445   ⊆ 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-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-nul 3986

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

This theorem is referenced by:  exmidn0m  4054  exmidsssn  4055  exmidpw  6704  exmidomni  6885  ss1oel2o  12610  exmidsbthrlem  12633