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Theorem pwtrufal 13384
 Description: A subset of the singleton cannot be anything other than or . Removing the double negation would change the meaning, as seen at exmid01 4130. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4129), 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

Proof of Theorem pwtrufal
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simprr 522 . . . . 5
2 simpll 519 . . . . . . . 8
3 simpl 108 . . . . . . . . . . . 12
43sselda 3103 . . . . . . . . . . 11
5 elsni 3551 . . . . . . . . . . 11
64, 5syl 14 . . . . . . . . . 10
7 simpr 109 . . . . . . . . . 10
86, 7eqeltrrd 2218 . . . . . . . . 9
98snssd 3674 . . . . . . . 8
102, 9eqssd 3120 . . . . . . 7
1110ex 114 . . . . . 6
1211exlimdv 1792 . . . . 5
131, 12mtod 653 . . . 4
14 notm0 3389 . . . 4
1513, 14sylib 121 . . 3
16 simprl 521 . . 3
1715, 16pm2.65da 651 . 2
18 ioran 742 . 2
1917, 18sylnibr 667 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 698   wceq 1332  wex 1469   wcel 1481   wss 3077  c0 3369  csn 3533 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-dif 3079  df-in 3083  df-ss 3090  df-nul 3370  df-sn 3539 This theorem is referenced by:  pwle2  13385
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