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Theorem ssdifin0 3304
 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.)
Assertion
Ref Expression
ssdifin0

Proof of Theorem ssdifin0
StepHypRef Expression
1 ssrin 3162 . 2
2 incom 3129 . . 3
3 disjdif 3296 . . 3
42, 3eqtri 2060 . 2
5 sseq0 3258 . 2
61, 4, 5sylancl 392 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243   cdif 2914   cin 2916   wss 2917  c0 3224 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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  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-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by:  ssdifeq0  3305
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