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Mirrors > Home > ILE Home > Th. List > dtruex | Unicode version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4115 can also be summarized as "at least two sets exist", the difference is that dtruarb 4115 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific , we can construct a set which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dtruex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . 5 | |
2 | 1 | snex 4109 | . . . 4 |
3 | 2 | isseti 2694 | . . 3 |
4 | elirrv 4463 | . . . . . . 7 | |
5 | vsnid 3557 | . . . . . . . 8 | |
6 | eleq2 2203 | . . . . . . . 8 | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 |
8 | 4, 7 | mto 651 | . . . . . 6 |
9 | eqtr 2157 | . . . . . 6 | |
10 | 8, 9 | mto 651 | . . . . 5 |
11 | ancom 264 | . . . . 5 | |
12 | 10, 11 | mtbi 659 | . . . 4 |
13 | 12 | imnani 680 | . . 3 |
14 | 3, 13 | eximii 1581 | . 2 |
15 | equcom 1682 | . . . 4 | |
16 | 15 | notbii 657 | . . 3 |
17 | 16 | exbii 1584 | . 2 |
18 | 14, 17 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1331 wex 1468 wcel 1480 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-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-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: dtru 4475 eunex 4476 brprcneu 5414 |
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