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Mirrors > Home > ILE Home > Th. List > dtru | Unicode version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4474. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dtru |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtruex 4474 | . 2 | |
2 | exnalim 1625 | . 2 | |
3 | 1, 2 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wal 1329 wex 1468 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 |
This theorem is referenced by: oprabidlem 5802 |
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