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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 4209 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 4209 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 2755 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | snex 4203 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
3 | 2 | isseti 2760 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | elirrv 4565 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() | |
5 | vsnid 3639 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eleq2 2253 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | mpbiri 168 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 4, 7 | mto 663 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | eqtr 2207 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | mto 663 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | ancom 266 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | mtbi 671 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 12 | imnani 692 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 3, 13 | eximii 1613 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | equcom 1717 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | notbii 669 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | exbii 1616 |
. 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 |
This theorem is referenced by: dtru 4577 eunex 4578 brprcneu 5527 |
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