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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 4462 | . . 3 | |
2 | eqrel 4429 | . . 3 | |
3 | 1, 2 | mpan2 401 | . 2 |
4 | eq0 3239 | . . 3 | |
5 | alnex 1388 | . . . . . 6 | |
6 | vex 2560 | . . . . . . 7 | |
7 | 6 | eldm2 4533 | . . . . . 6 |
8 | 5, 7 | xchbinxr 608 | . . . . 5 |
9 | noel 3228 | . . . . . . 7 | |
10 | 9 | nbn 615 | . . . . . 6 |
11 | 10 | albii 1359 | . . . . 5 |
12 | 8, 11 | bitr3i 175 | . . . 4 |
13 | 12 | albii 1359 | . . 3 |
14 | 4, 13 | bitr2i 174 | . 2 |
15 | 3, 14 | syl6bb 185 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 c0 3224 cop 3378 cdm 4345 wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-dm 4355 |
This theorem is referenced by: relrn0 4594 fnresdisj 5009 fn0 5018 fsnunfv 5363 |
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