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Mirrors > Home > ILE Home > Th. List > reldm0 | Unicode version |
Description: A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
reldm0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4634 | . . 3 | |
2 | eqrel 4598 | . . 3 | |
3 | 1, 2 | mpan2 421 | . 2 |
4 | eq0 3351 | . . 3 | |
5 | alnex 1460 | . . . . . 6 | |
6 | vex 2663 | . . . . . . 7 | |
7 | 6 | eldm2 4707 | . . . . . 6 |
8 | 5, 7 | xchbinxr 657 | . . . . 5 |
9 | noel 3337 | . . . . . . 7 | |
10 | 9 | nbn 673 | . . . . . 6 |
11 | 10 | albii 1431 | . . . . 5 |
12 | 8, 11 | bitr3i 185 | . . . 4 |
13 | 12 | albii 1431 | . . 3 |
14 | 4, 13 | bitr2i 184 | . 2 |
15 | 3, 14 | syl6bb 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 c0 3333 cop 3500 cdm 4509 wrel 4514 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-dm 4519 |
This theorem is referenced by: relrn0 4771 fnresdisj 5203 fn0 5212 fsnunfv 5589 setsresg 11908 metn0 12458 |
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