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Theorem ssdifin0 4192
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 3986 . 2 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶))
2 incom 3956 . . 3 ((𝐵𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵𝐶))
3 disjdif 4182 . . 3 (𝐶 ∩ (𝐵𝐶)) = ∅
42, 3eqtri 2793 . 2 ((𝐵𝐶) ∩ 𝐶) = ∅
5 sseq0 4119 . 2 (((𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶) ∧ ((𝐵𝐶) ∩ 𝐶) = ∅) → (𝐴𝐶) = ∅)
61, 4, 5sylancl 574 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  cdif 3720  cin 3722  wss 3723  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064
This theorem is referenced by:  ssdifeq0  4193  marypha1lem  8495  numacn  9072  mreexexlem2d  16513  mreexexlem4d  16515  nrmsep2  21381  isnrm3  21384
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