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Mirrors > Home > ILE Home > Th. List > Mathboxes > pwtrufal | Unicode version |
Description: A subset of the singleton cannot be anything other than or . Removing the double negation would change the meaning, as seen at exmid01 4171. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4169), 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.) |
Ref | Expression |
---|---|
pwtrufal |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 522 | . . . . 5 | |
2 | simpll 519 | . . . . . . . 8 | |
3 | simpl 108 | . . . . . . . . . . . 12 | |
4 | 3 | sselda 3137 | . . . . . . . . . . 11 |
5 | elsni 3588 | . . . . . . . . . . 11 | |
6 | 4, 5 | syl 14 | . . . . . . . . . 10 |
7 | simpr 109 | . . . . . . . . . 10 | |
8 | 6, 7 | eqeltrrd 2242 | . . . . . . . . 9 |
9 | 8 | snssd 3712 | . . . . . . . 8 |
10 | 2, 9 | eqssd 3154 | . . . . . . 7 |
11 | 10 | ex 114 | . . . . . 6 |
12 | 11 | exlimdv 1806 | . . . . 5 |
13 | 1, 12 | mtod 653 | . . . 4 |
14 | notm0 3424 | . . . 4 | |
15 | 13, 14 | sylib 121 | . . 3 |
16 | simprl 521 | . . 3 | |
17 | 15, 16 | pm2.65da 651 | . 2 |
18 | ioran 742 | . 2 | |
19 | 17, 18 | sylnibr 667 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wceq 1342 wex 1479 wcel 2135 wss 3111 c0 3404 csn 3570 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 |
This theorem is referenced by: pwle2 13719 |
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