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  8492  numacn  9962  ramub1lem1  16988  ramub1lem2  16989  mreexexlem2d  17602  mreexexlem3d  17603  mreexexlem4d  17604  acsfiindd  18510  dpjidcl  20026  clsval2  23025  llycmpkgen2  23525  1stckgen  23529  alexsublem  24019  bcthlem3  25303  pmtrcnelor  33167  lfuhgr  35316  neibastop2lem  36558  pibt2  37747  eldioph2lem2  43207  limccog  46068  fourierdlem56  46608  fourierdlem95  46647