Theorem difss2d 4078

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4077. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
difss2d.1	⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))

Assertion

Ref	Expression
difss2d	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem difss2d

Step	Hyp	Ref	Expression
1		difss2d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))
2		difss2 4077	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ∖ cdif 4048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050

This theorem is used by:  oacomf1olem  8603  numacn  10158  ramub1lem1  17236  ramub1lem2  17237  mreexexlem2d  17851  mreexexlem3d  17852  mreexexlem4d  17853  acsfiindd  18759  dpjidcl  20307  clsval2  23401  llycmpkgen2  23902  1stckgen  23906  alexsublem  24396  bcthlem3  25680  lfuhgr  29759  pmtrcnelor  33728  neibastop2lem  37339  pibt2  38545  eldioph2lem2  43976  limccog  46847  fourierdlem56  47387  fourierdlem95  47426