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Theorem pwtrufal 13245
Description: A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4121. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4120), 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 521 . . . . 5 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = {∅})
2 simpll 518 . . . . . . . 8 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝐴 ⊆ {∅})
3 simpl 108 . . . . . . . . . . . 12 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 ⊆ {∅})
43sselda 3097 . . . . . . . . . . 11 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥 ∈ {∅})
5 elsni 3545 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
64, 5syl 14 . . . . . . . . . 10 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥 = ∅)
7 simpr 109 . . . . . . . . . 10 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝑥𝐴)
86, 7eqeltrrd 2217 . . . . . . . . 9 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → ∅ ∈ 𝐴)
98snssd 3665 . . . . . . . 8 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → {∅} ⊆ 𝐴)
102, 9eqssd 3114 . . . . . . 7 (((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) ∧ 𝑥𝐴) → 𝐴 = {∅})
1110ex 114 . . . . . 6 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (𝑥𝐴𝐴 = {∅}))
1211exlimdv 1791 . . . . 5 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → (∃𝑥 𝑥𝐴𝐴 = {∅}))
131, 12mtod 652 . . . 4 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ ∃𝑥 𝑥𝐴)
14 notm0 3383 . . . 4 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
1513, 14sylib 121 . . 3 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → 𝐴 = ∅)
16 simprl 520 . . 3 ((𝐴 ⊆ {∅} ∧ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅})) → ¬ 𝐴 = ∅)
1715, 16pm2.65da 650 . 2 (𝐴 ⊆ {∅} → ¬ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
18 ioran 741 . 2 (¬ (𝐴 = ∅ ∨ 𝐴 = {∅}) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐴 = {∅}))
1917, 18sylnibr 666 1 (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697   = wceq 1331  wex 1468  wcel 1480  wss 3071  c0 3363  {csn 3527
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533
This theorem is referenced by:  pwle2  13246
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