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Theorem ssdifin0 4392
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 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)

Proof of Theorem ssdifin0
StepHypRef Expression
1 ssrin 4163 . 2 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶))
2 incom 4131 . . 3 ((𝐵𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵𝐶))
3 disjdif 4382 . . 3 (𝐶 ∩ (𝐵𝐶)) = ∅
42, 3eqtri 2824 . 2 ((𝐵𝐶) ∩ 𝐶) = ∅
5 sseq0 4310 . 2 (((𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶) ∧ ((𝐵𝐶) ∩ 𝐶) = ∅) → (𝐴𝐶) = ∅)
61, 4, 5sylancl 589 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cdif 3881  cin 3883  wss 3884  c0 4246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247
This theorem is referenced by:  ssdifeq0  4393  marypha1lem  8885  numacn  9464  mreexexlem2d  16911  mreexexlem4d  16913  nrmsep2  21964  isnrm3  21967
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