| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > dmrnssfld | Unicode version | ||
| Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.) |
| Ref | Expression |
|---|---|
| dmrnssfld |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 2775 |
. . . . 5
| |
| 2 | 1 | eldm2 4876 |
. . . 4
|
| 3 | 1 | prid1 3739 |
. . . . . 6
|
| 4 | vex 2775 |
. . . . . . . . . 10
| |
| 5 | 1, 4 | uniop 4300 |
. . . . . . . . 9
|
| 6 | 1, 4 | uniopel 4301 |
. . . . . . . . 9
|
| 7 | 5, 6 | eqeltrrid 2293 |
. . . . . . . 8
|
| 8 | elssuni 3878 |
. . . . . . . 8
| |
| 9 | 7, 8 | syl 14 |
. . . . . . 7
|
| 10 | 9 | sseld 3192 |
. . . . . 6
|
| 11 | 3, 10 | mpi 15 |
. . . . 5
|
| 12 | 11 | exlimiv 1621 |
. . . 4
|
| 13 | 2, 12 | sylbi 121 |
. . 3
|
| 14 | 13 | ssriv 3197 |
. 2
|
| 15 | 4 | elrn2 4920 |
. . . 4
|
| 16 | 4 | prid2 3740 |
. . . . . 6
|
| 17 | 9 | sseld 3192 |
. . . . . 6
|
| 18 | 16, 17 | mpi 15 |
. . . . 5
|
| 19 | 18 | exlimiv 1621 |
. . . 4
|
| 20 | 15, 19 | sylbi 121 |
. . 3
|
| 21 | 20 | ssriv 3197 |
. 2
|
| 22 | 14, 21 | unssi 3348 |
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-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-cnv 4683 df-dm 4685 df-rn 4686 |
| This theorem is referenced by: dmexg 4942 rnexg 4943 relfld 5211 relcoi2 5213 |
| Copyright terms: Public domain | W3C validator |