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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 4480 | . . 3 | |
2 | eqrel 4447 | . . 3 | |
3 | 1, 2 | mpan2 415 | . 2 |
4 | eq0 3266 | . . 3 | |
5 | alnex 1428 | . . . . . 6 | |
6 | vex 2604 | . . . . . . 7 | |
7 | 6 | eldm2 4551 | . . . . . 6 |
8 | 5, 7 | xchbinxr 640 | . . . . 5 |
9 | noel 3255 | . . . . . . 7 | |
10 | 9 | nbn 647 | . . . . . 6 |
11 | 10 | albii 1399 | . . . . 5 |
12 | 8, 11 | bitr3i 184 | . . . 4 |
13 | 12 | albii 1399 | . . 3 |
14 | 4, 13 | bitr2i 183 | . 2 |
15 | 3, 14 | syl6bb 194 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 103 wal 1282 wceq 1284 wex 1421 wcel 1433 c0 3251 cop 3401 cdm 4363 wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-dm 4373 |
This theorem is referenced by: relrn0 4612 fnresdisj 5029 fn0 5038 fsnunfv 5384 |
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