Theorem ssindif0im 3303

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 3202	. . 3 ⊢ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))
2		sstr 3007	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))) → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
3	1, 2	mpan2 415	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
4		disj2 3299	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
5	3, 4	sylibr 132	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1284  Vcvv 2601   ∖ cdif 2970   ∩ cin 2972   ⊆ wss 2973  ∅c0 3251

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by: (None)