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 4140	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	sseq2i 4012	. 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵)
4	1, 3	bitr2i 276	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1539  Vcvv 3479   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333

This theorem is referenced by:  setind  9775