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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 4182. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4180), 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 527 | . . . . 5 | |
2 | simpll 524 | . . . . . . . 8 | |
3 | simpl 108 | . . . . . . . . . . . 12 | |
4 | 3 | sselda 3147 | . . . . . . . . . . 11 |
5 | elsni 3599 | . . . . . . . . . . 11 | |
6 | 4, 5 | syl 14 | . . . . . . . . . 10 |
7 | simpr 109 | . . . . . . . . . 10 | |
8 | 6, 7 | eqeltrrd 2248 | . . . . . . . . 9 |
9 | 8 | snssd 3723 | . . . . . . . 8 |
10 | 2, 9 | eqssd 3164 | . . . . . . 7 |
11 | 10 | ex 114 | . . . . . 6 |
12 | 11 | exlimdv 1812 | . . . . 5 |
13 | 1, 12 | mtod 658 | . . . 4 |
14 | notm0 3434 | . . . 4 | |
15 | 13, 14 | sylib 121 | . . 3 |
16 | simprl 526 | . . 3 | |
17 | 15, 16 | pm2.65da 656 | . 2 |
18 | ioran 747 | . 2 | |
19 | 17, 18 | sylnibr 672 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wceq 1348 wex 1485 wcel 2141 wss 3121 c0 3414 csn 3581 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3587 |
This theorem is referenced by: pwle2 13953 |
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