Theorem ssdifin0 4433

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 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)

Proof of Theorem ssdifin0

Step	Hyp	Ref	Expression
1		ssrin 4212	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶))
2		incom 4180	. . 3 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵 ∖ 𝐶))
3		disjdif 4423	. . 3 ⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅
4	2, 3	eqtri 2846	. 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅
5		sseq0 4355	. 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)
6	1, 4, 5	sylancl 588	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294

This theorem is referenced by:  ssdifeq0  4434  marypha1lem  8899  numacn  9477  mreexexlem2d  16918  mreexexlem4d  16920  nrmsep2  21966  isnrm3  21969