Theorem difss2d 4079

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4078. (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 4078	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3886   ⊆ wss 3889

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-ss 3906

This theorem is referenced by:  oacomf1olem  8499  numacn  9971  ramub1lem1  16997  ramub1lem2  16998  mreexexlem2d  17611  mreexexlem3d  17612  mreexexlem4d  17613  acsfiindd  18519  dpjidcl  20035  clsval2  23015  llycmpkgen2  23515  1stckgen  23519  alexsublem  24009  bcthlem3  25293  pmtrcnelor  33152  lfuhgr  35300  neibastop2lem  36542  pibt2  37733  eldioph2lem2  43193  limccog  46050  fourierdlem56  46590  fourierdlem95  46629