Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4302. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dtru | ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtruex 4302 | . 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | |
2 | exnalim 1577 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1282 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 |
This theorem is referenced by: oprabidlem 5556 |
Copyright terms: Public domain | W3C validator |