Theorem exmid01 4231

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 4228	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2		df-dc 836	. . . . 5 ⊢ (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3		orcom 729	. . . . . 6 ⊢ ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4		simpll 527	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ {∅})
5		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
6	4, 5	sseldd 3184	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ {∅})
7		velsn 3639	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
8	6, 7	sylib 122	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
9	8, 5	eqeltrrd 2274	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥)
10		simplr 528	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → ¬ ∅ ∈ 𝑥)
11	9, 10	pm2.65da 662	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦 ∈ 𝑥)
12	11	eq0rdv 3495	. . . . . . . . 9 ⊢ ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
13	12	ex 115	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
14		noel 3454	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ∅
15		eleq2 2260	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
16	14, 15	mtbiri 676	. . . . . . . 8 ⊢ (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
17	13, 16	impbid1 142	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥 ↔ 𝑥 = ∅))
18		ss1o0el1 4230	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (∅ ∈ 𝑥 ↔ 𝑥 = {∅}))
19	17, 18	orbi12d 794	. . . . . 6 ⊢ (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
20	3, 19	bitrid 192	. . . . 5 ⊢ (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
21	2, 20	bitrid 192	. . . 4 ⊢ (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
22	21	pm5.74i 180	. . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
23	22	albii 1484	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
24	1, 23	bitri 184	1 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835  ∀wal 1362   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  ∅c0 3450  {csn 3622  EXMIDwem 4227

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-exmid 4228

This theorem is referenced by:  exmid1dc  4233  exmidn0m  4234  exmidsssn  4235  exmidpw  6969  exmidpweq  6970  exmidomni  7208  ss1oel2o  15638  exmidsbthrlem  15666  sbthom  15670