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Theorem ssdifin0 4022
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 3816 . 2 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶))
2 incom 3783 . . 3 ((𝐵𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵𝐶))
3 disjdif 4012 . . 3 (𝐶 ∩ (𝐵𝐶)) = ∅
42, 3eqtri 2643 . 2 ((𝐵𝐶) ∩ 𝐶) = ∅
5 sseq0 3947 . 2 (((𝐴𝐶) ⊆ ((𝐵𝐶) ∩ 𝐶) ∧ ((𝐵𝐶) ∩ 𝐶) = ∅) → (𝐴𝐶) = ∅)
61, 4, 5sylancl 693 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cdif 3552  cin 3554  wss 3555  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892
This theorem is referenced by:  ssdifeq0  4023  marypha1lem  8283  numacn  8816  mreexexlem2d  16226  mreexexlem4d  16228  nrmsep2  21070  isnrm3  21073
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