Theorem ssindif0im 3551

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 3442	. . 3 ⊢ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))
2		sstr 3232	. . 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 3547	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
5	3, 4	sylibr 134	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2799   ∖ cdif 3194   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by: (None)