Theorem ssindif0 4392

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 4386	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵)))
2		ddif 4071	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	sseq2i 3944	. 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵)
4	1, 3	bitr2i 277	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Syntax hints:   ↔ wb 207   = wceq 1547  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262

This theorem is referenced by:  setind  9659  setindregs  35311