Theorem pwtrufal 14030

Description: A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4184. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4182), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)

Assertion

Ref	Expression
pwtrufal	⊢ (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))

Proof of Theorem pwtrufal

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 527	. . . . 5 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = {∅})
2		simpll 524	. . . . . . . 8 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ {∅})
3		simpl 108	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 ⊆ {∅})
4	3	sselda 3147	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
5		elsni 3601	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
6	4, 5	syl 14	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
7		simpr 109	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	6, 7	eqeltrrd 2248	. . . . . . . . 9 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
9	8	snssd 3725	. . . . . . . 8 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
10	2, 9	eqssd 3164	. . . . . . 7 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
11	10	ex 114	. . . . . 6 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (𝑥 ∈ 𝐴 → 𝐴 = {∅}))
12	11	exlimdv 1812	. . . . 5 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {∅}))
13	1, 12	mtod 658	. . . 4 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
14		notm0 3435	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
15	13, 14	sylib 121	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 = ∅)
16		simprl 526	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = ∅)
17	15, 16	pm2.65da 656	. 2 ⊢ (𝐴 ⊆ {∅} → ¬ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
18		ioran 747	. 2 ⊢ (¬ (𝐴 = ∅ ∨ 𝐴 = {∅}) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
19	17, 18	sylnibr 672	1 ⊢ (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ⊆ wss 3121  ∅c0 3414  {csn 3583

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589

This theorem is referenced by:  pwle2  14031