Theorem ssindif0 4463

Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Proof of Theorem ssindif0

Step	Hyp	Ref	Expression
1		disj2 4457	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2		ddif 4136	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	sseq2i 4011	. 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵)
4	1, 3	bitr2i 276	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   ↔ wb 205   = wceq 1540  Vcvv 3473   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  setind  9735