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Mirrors > Home > ILE Home > Th. List > dmsn0el | Unicode version |
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Ref | Expression |
---|---|
dmsn0el |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelelxp 4538 | . . . . 5 | |
2 | 1 | con2i 601 | . . . 4 |
3 | dmsnm 4974 | . . . 4 | |
4 | 2, 3 | sylnib 650 | . . 3 |
5 | alnex 1460 | . . 3 | |
6 | 4, 5 | sylibr 133 | . 2 |
7 | eq0 3351 | . 2 | |
8 | 6, 7 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wal 1314 wceq 1316 wex 1453 wcel 1465 cvv 2660 c0 3333 csn 3497 cxp 4507 cdm 4509 |
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-ne 2286 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-dm 4519 |
This theorem is referenced by: (None) |
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