Theorem ssdif2d 3139

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

Hypotheses

Ref	Expression
ssdifd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ssdif2d.2	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Assertion

Ref	Expression
ssdif2d	⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))

Proof of Theorem ssdif2d

Step	Hyp	Ref	Expression
1		ssdif2d.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
2	1	sscond 3137	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶))
3		ssdifd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	ssdifd 3136	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
5	2, 4	sstrd 3035	1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:   → wi 4   ∖ cdif 2996   ⊆ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012

This theorem is referenced by: (None)