Theorem difin0 4497

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 4259	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		ssdif0 4389	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅)
3	1, 2	mpbi 230	1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅

Syntax hints:   = wceq 1537   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-nul 4353

This theorem is referenced by:  volinun  25602