Theorem difin0 4427

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 4134	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		ssdif0 4313	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅)
3	1, 2	mpbi 233	1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279

This theorem is used by:  volinun  25868