Theorem ssindif0im 3497

Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0im	⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Proof of Theorem ssindif0im

Step	Hyp	Ref	Expression
1		ddifss 3388	. . 3 ⊢ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))
2		sstr 3178	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))) → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
4		disj2 3493	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
5	3, 4	sylibr 134	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2752   ∖ cdif 3141   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438

This theorem is referenced by: (None)