Theorem inindif 4324

Description: The intersection and class difference of a class with another class are disjoint. With inundif 4435, 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 4183	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		ssinss1 4191	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶)
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶
4		inssdif0 4322	. 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 1570   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280

This theorem is used by:  resf1o  33230  indsumin  33336  gsummptres  33521  measunl  34757  carsgclctun  34862  probdif  34961  hgt750lemd  35186  redvmptabs  43249