Theorem ssindif0 4416

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 4410	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2		ddif 4093	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	sseq2i 3963	. 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵)
4	1, 3	bitr2i 276	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-dif 3904  df-in 3908  df-ss 3918  df-nul 4286

This theorem is referenced by:  setind  9656  setindregs  35286