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Mirrors > Home > ILE Home > Th. List > dtru | GIF 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 4482. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dtru | ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtruex 4482 | . 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | |
2 | exnalim 1626 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1330 ∃wex 1469 |
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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 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-ne 2310 df-ral 2422 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 |
This theorem is referenced by: oprabidlem 5810 |
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