Theorem ssdifss 2171

Description: Preservation of a subclass relationship by class difference.

Assertion

Ref	Expression
ssdifss	⊢ (A ⊆ B → (A ∖ C) ⊆ B)

Proof of Theorem ssdifss

Step	Hyp	Ref	Expression
1		difss 2170	. 2 ⊢ (A ∖ C) ⊆ A
2		sstr 2075	. 2 ⊢ (((A ∖ C) ⊆ A ⋀ A ⊆ B) → (A ∖ C) ⊆ B)
3	1, 2	mpan 697	1 ⊢ (A ⊆ B → (A ∖ C) ⊆ B)

Syntax hints:   → wi 3   ∖ cdif 2047   ⊆ wss 2050

This theorem is referenced by:  unblem1 4551  xrsupss 6080  xrinfmss 6081  islp2 7744  rcfpfillem6 10568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056

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