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Mirrors > Home > ILE Home > Th. List > ssdifin0 | Unicode version |
Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
ssdifin0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 3332 | . 2 | |
2 | incom 3299 | . . 3 | |
3 | disjdif 3466 | . . 3 | |
4 | 2, 3 | eqtri 2178 | . 2 |
5 | sseq0 3435 | . 2 | |
6 | 1, 4, 5 | sylancl 410 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 cdif 3099 cin 3101 wss 3102 c0 3394 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 |
This theorem is referenced by: ssdifeq0 3476 |
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