Theorem difss2d 4135

Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4134. (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 4134	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3946   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966

This theorem is referenced by:  oacomf1olem  8564  numacn  10044  ramub1lem1  16959  ramub1lem2  16960  mreexexlem2d  17589  mreexexlem3d  17590  mreexexlem4d  17591  acsfiindd  18506  dpjidcl  19928  clsval2  22554  llycmpkgen2  23054  1stckgen  23058  alexsublem  23548  bcthlem3  24843  pmtrcnelor  32252  lfuhgr  34108  neibastop2lem  35245  pibt2  36298  eldioph2lem2  41499  limccog  44336  fourierdlem56  44878  fourierdlem95  44917