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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 4154 can also be summarized as "at least two sets exist", the difference is that dtruarb 4154 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 2715 | . . . . 5 | |
2 | 1 | snex 4148 | . . . 4 |
3 | 2 | isseti 2720 | . . 3 |
4 | elirrv 4509 | . . . . . . 7 | |
5 | vsnid 3593 | . . . . . . . 8 | |
6 | eleq2 2221 | . . . . . . . 8 | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 |
8 | 4, 7 | mto 652 | . . . . . 6 |
9 | eqtr 2175 | . . . . . 6 | |
10 | 8, 9 | mto 652 | . . . . 5 |
11 | ancom 264 | . . . . 5 | |
12 | 10, 11 | mtbi 660 | . . . 4 |
13 | 12 | imnani 681 | . . 3 |
14 | 3, 13 | eximii 1582 | . 2 |
15 | equcom 1686 | . . . 4 | |
16 | 15 | notbii 658 | . . 3 |
17 | 16 | exbii 1585 | . 2 |
18 | 14, 17 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1335 wex 1472 wcel 2128 csn 3561 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-setind 4498 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 |
This theorem is referenced by: dtru 4521 eunex 4522 brprcneu 5463 |
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