Theorem difss2d 4080

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4079. (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 4079	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3887   ⊆ wss 3890

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 3432  df-dif 3893  df-ss 3907

This theorem is referenced by:  oacomf1olem  8490  numacn  9960  ramub1lem1  16986  ramub1lem2  16987  mreexexlem2d  17600  mreexexlem3d  17601  mreexexlem4d  17602  acsfiindd  18508  dpjidcl  20024  clsval2  23024  llycmpkgen2  23524  1stckgen  23528  alexsublem  24018  bcthlem3  25302  pmtrcnelor  33172  lfuhgr  35321  neibastop2lem  36563  pibt2  37744  eldioph2lem2  43204  limccog  46065  fourierdlem56  46605  fourierdlem95  46644