Theorem difin0 4434

Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difin0	⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅

Proof of Theorem difin0

Step	Hyp	Ref	Expression
1		inss2 4189	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		ssdif0 4320	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅)
3	1, 2	mpbi 233	1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅

Syntax hints:   = wceq 1568   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286

This theorem is referenced by:  volinun  25684