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Theorem ssdifin0 4244
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 4033 . 2 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶))
2 incom 4003 . . 3 ((𝐵𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵𝐶))
3 disjdif 4234 . . 3 (𝐶 ∩ (𝐵𝐶)) = ∅
42, 3eqtri 2821 . 2 ((𝐵𝐶) ∩ 𝐶) = ∅
5 sseq0 4171 . 2 (((𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶) ∧ ((𝐵𝐶) ∩ 𝐶) = ∅) → (𝐴𝐶) = ∅)
61, 4, 5sylancl 581 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  cdif 3766  cin 3768  wss 3769  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116
This theorem is referenced by:  ssdifeq0  4245  marypha1lem  8581  numacn  9158  mreexexlem2d  16620  mreexexlem4d  16622  nrmsep2  21489  isnrm3  21492
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