Theorem difin0 3624

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 3477	. 2 ⊢ (A ∩ B) ⊆ B
2		ssdif0 3610	. 2 ⊢ ((A ∩ B) ⊆ B ↔ ((A ∩ B) ∖ B) = ∅)
3	1, 2	mpbi 199	1 ⊢ ((A ∩ B) ∖ B) = ∅

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)