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Theorem pwtrufal 14717
Description: A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4198. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4196), 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.
StepHypRef Expression
1 simprr 531 . . . . 5 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = {∅})
2 simpll 527 . . . . . . . 8 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝐴 ⊆ {∅})
3 simpl 109 . . . . . . . . . . . 12 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 ⊆ {∅})
43sselda 3155 . . . . . . . . . . 11 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥 ∈ {∅})
5 elsni 3610 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
64, 5syl 14 . . . . . . . . . 10 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥 = ∅)
7 simpr 110 . . . . . . . . . 10 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥𝐴)
86, 7eqeltrrd 2255 . . . . . . . . 9 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → ∅ ∈ 𝐴)
98snssd 3737 . . . . . . . 8 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → {∅} ⊆ 𝐴)
102, 9eqssd 3172 . . . . . . 7 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝐴 = {∅})
1110ex 115 . . . . . 6 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (𝑥𝐴𝐴 = {∅}))
1211exlimdv 1819 . . . . 5 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (∃𝑥 𝑥𝐴𝐴 = {∅}))
131, 12mtod 663 . . . 4 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ ∃𝑥 𝑥𝐴)
14 notm0 3443 . . . 4 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
1513, 14sylib 122 . . 3 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 = ∅)
16 simprl 529 . . 3 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = ∅)
1715, 16pm2.65da 661 . 2 (𝐴 ⊆ {∅} → ¬ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
18 ioran 752 . 2 (¬ (𝐴 = ∅ ∨ 𝐴 = {∅}) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
1917, 18sylnibr 677 1 (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wex 1492  wcel 2148  wss 3129  c0 3422  {csn 3592
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598
This theorem is referenced by:  pwle2  14718
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