Theorem difss2d 4093

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4092. (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 4092	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3900   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920

This theorem is referenced by:  oacomf1olem  8501  numacn  9971  ramub1lem1  16966  ramub1lem2  16967  mreexexlem2d  17580  mreexexlem3d  17581  mreexexlem4d  17582  acsfiindd  18488  dpjidcl  20001  clsval2  23006  llycmpkgen2  23506  1stckgen  23510  alexsublem  24000  bcthlem3  25294  pmtrcnelor  33184  lfuhgr  35331  neibastop2lem  36573  pibt2  37669  eldioph2lem2  43115  limccog  45977  fourierdlem56  46517  fourierdlem95  46556