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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 4664 | . . 3 | |
2 | eqrel 4628 | . . 3 | |
3 | 1, 2 | mpan2 421 | . 2 |
4 | eq0 3381 | . . 3 | |
5 | alnex 1475 | . . . . . 6 | |
6 | vex 2689 | . . . . . . 7 | |
7 | 6 | eldm2 4737 | . . . . . 6 |
8 | 5, 7 | xchbinxr 672 | . . . . 5 |
9 | noel 3367 | . . . . . . 7 | |
10 | 9 | nbn 688 | . . . . . 6 |
11 | 10 | albii 1446 | . . . . 5 |
12 | 8, 11 | bitr3i 185 | . . . 4 |
13 | 12 | albii 1446 | . . 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 1329 wceq 1331 wex 1468 wcel 1480 c0 3363 cop 3530 cdm 4539 wrel 4544 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 |
This theorem is referenced by: relrn0 4801 fnresdisj 5233 fn0 5242 fsnunfv 5621 setsresg 12007 metn0 12557 |
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