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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 4177 can also be summarized as "at least two sets exist", the difference is that dtruarb 4177 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 2733 | . . . . 5 | |
2 | 1 | snex 4171 | . . . 4 |
3 | 2 | isseti 2738 | . . 3 |
4 | elirrv 4532 | . . . . . . 7 | |
5 | vsnid 3615 | . . . . . . . 8 | |
6 | eleq2 2234 | . . . . . . . 8 | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 |
8 | 4, 7 | mto 657 | . . . . . 6 |
9 | eqtr 2188 | . . . . . 6 | |
10 | 8, 9 | mto 657 | . . . . 5 |
11 | ancom 264 | . . . . 5 | |
12 | 10, 11 | mtbi 665 | . . . 4 |
13 | 12 | imnani 686 | . . 3 |
14 | 3, 13 | eximii 1595 | . 2 |
15 | equcom 1699 | . . . 4 | |
16 | 15 | notbii 663 | . . 3 |
17 | 16 | exbii 1598 | . 2 |
18 | 14, 17 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1348 wex 1485 wcel 2141 csn 3583 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 |
This theorem is referenced by: dtru 4544 eunex 4545 brprcneu 5489 |
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