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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 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.) |
Ref | Expression |
---|---|
pwtrufal |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 521 | . . . . 5 | |
2 | simpll 518 | . . . . . . . 8 | |
3 | simpl 108 | . . . . . . . . . . . 12 | |
4 | 3 | sselda 3097 | . . . . . . . . . . 11 |
5 | elsni 3545 | . . . . . . . . . . 11 | |
6 | 4, 5 | syl 14 | . . . . . . . . . 10 |
7 | simpr 109 | . . . . . . . . . 10 | |
8 | 6, 7 | eqeltrrd 2217 | . . . . . . . . 9 |
9 | 8 | snssd 3665 | . . . . . . . 8 |
10 | 2, 9 | eqssd 3114 | . . . . . . 7 |
11 | 10 | ex 114 | . . . . . 6 |
12 | 11 | exlimdv 1791 | . . . . 5 |
13 | 1, 12 | mtod 652 | . . . 4 |
14 | notm0 3383 | . . . 4 | |
15 | 13, 14 | sylib 121 | . . 3 |
16 | simprl 520 | . . 3 | |
17 | 15, 16 | pm2.65da 650 | . 2 |
18 | ioran 741 | . 2 | |
19 | 17, 18 | sylnibr 666 | 1 |
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 13193 |
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