Theorem pwtrufal 16598

Description: A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4288. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4286), 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 533	. . . . 5 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = {∅})
2		simpll 527	. . . . . . . 8 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ {∅})
3		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 ⊆ {∅})
4	3	sselda 3227	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
5		elsni 3687	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
6	4, 5	syl 14	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
7		simpr 110	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	6, 7	eqeltrrd 2309	. . . . . . . . 9 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
9	8	snssd 3818	. . . . . . . 8 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
10	2, 9	eqssd 3244	. . . . . . 7 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
11	10	ex 115	. . . . . 6 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (𝑥 ∈ 𝐴 → 𝐴 = {∅}))
12	11	exlimdv 1867	. . . . 5 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {∅}))
13	1, 12	mtod 669	. . . 4 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
14		notm0 3515	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
15	13, 14	sylib 122	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 = ∅)
16		simprl 531	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = ∅)
17	15, 16	pm2.65da 667	. 2 ⊢ (𝐴 ⊆ {∅} → ¬ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
18		ioran 759	. 2 ⊢ (¬ (𝐴 = ∅ ∨ 𝐴 = {∅}) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
19	17, 18	sylnibr 683	1 ⊢ (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ⊆ wss 3200  ∅c0 3494  {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675

This theorem is referenced by:  pwle2  16599