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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 4191 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 4191 shows the existence of two sets which are not
equal to each
other, but this theorem says that given a specific ![]() ![]() |
Ref | Expression |
---|---|
dtruex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2740 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | snex 4185 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
3 | 2 | isseti 2745 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | elirrv 4547 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() | |
5 | vsnid 3624 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eleq2 2241 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | mpbiri 168 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 4, 7 | mto 662 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | eqtr 2195 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | mto 662 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | ancom 266 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | mtbi 670 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 12 | imnani 691 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 3, 13 | eximii 1602 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | equcom 1706 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | notbii 668 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | exbii 1605 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 14, 17 | mpbi 145 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 |
This theorem is referenced by: dtru 4559 eunex 4560 brprcneu 5508 |
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