Theorem ssdifssd 3311

Description: If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 3303. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssdifssd	⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Proof of Theorem ssdifssd

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssdifss 3303	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3163   ⊆ wss 3166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179

This theorem is referenced by:  zfz1isolem1  10985