Theorem inindif 4327

Description: The intersection and class difference of a class with another class are disjoint. With inundif 4438, this shows that such intersection and class difference partition the class 𝐴. (Contributed by Thierry Arnoux, 13-Sep-2017.)

Assertion

Ref	Expression
inindif	⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅

Proof of Theorem inindif

Step	Hyp	Ref	Expression
1		inss2 4137	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		ssinss1 4145	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶)
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶
4		inssdif0 4325	. 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
5	3, 4	mpbi 233	1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282

This theorem is used by:  resf1o  33378  indsumin  33484  gsummptres  33669  measunl  34905  carsgclctun  35009  probdif  35108  hgt750lemd  35333  redvmptabs  43557