Theorem pwtrufal 15642

Description: A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4231. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4229), 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 531	. . . . 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 3183	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
5		elsni 3640	. . . . . . . . . . 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 2274	. . . . . . . . 9 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
9	8	snssd 3767	. . . . . . . 8 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
10	2, 9	eqssd 3200	. . . . . . 7 ⊢ (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
11	10	ex 115	. . . . . 6 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (𝑥 ∈ 𝐴 → 𝐴 = {∅}))
12	11	exlimdv 1833	. . . . 5 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {∅}))
13	1, 12	mtod 664	. . . 4 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
14		notm0 3471	. . . 4 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
15	13, 14	sylib 122	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 = ∅)
16		simprl 529	. . 3 ⊢ ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = ∅)
17	15, 16	pm2.65da 662	. 2 ⊢ (𝐴 ⊆ {∅} → ¬ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
18		ioran 753	. 2 ⊢ (¬ (𝐴 = ∅ ∨ 𝐴 = {∅}) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
19	17, 18	sylnibr 678	1 ⊢ (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167   ⊆ wss 3157  ∅c0 3450  {csn 3622

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

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  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

This theorem is referenced by:  pwle2  15643