Theorem difss2d 4089

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4088. (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 4088	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∖ cdif 3899   ⊆ wss 3902

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-ss 3919

This theorem is used by:  oacomf1olem  8555  numacn  10056  ramub1lem1  17124  ramub1lem2  17125  mreexexlem2d  17739  mreexexlem3d  17740  mreexexlem4d  17741  acsfiindd  18647  dpjidcl  20193  clsval2  23281  llycmpkgen2  23782  1stckgen  23786  alexsublem  24276  bcthlem3  25560  lfuhgr  29613  pmtrcnelor  33539  neibastop2lem  36987  pibt2  38179  eldioph2lem2  43614  limccog  46458  fourierdlem56  46998  fourierdlem95  47037