Theorem ssindif0 4415

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 4409	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2		ddif 4092	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	sseq2i 3963	. 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵)
4	1, 3	bitr2i 278	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   ↔ wb 208   = wceq 1559  Vcvv 3453   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4284

This theorem is referenced by:  setind  9696  setindregs  35387