Theorem inindif 4341

Description: The intersection and class difference of a class with another class are disjoint. With inundif 4445, 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 4204	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		ssinss1 4212	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶)
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶
4		inssdif0 4340	. 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
5	3, 4	mpbi 230	1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300

This theorem is referenced by:  resf1o  32660  indsumin  32792  gsummptres  32999  measunl  34213  carsgclctun  34319  probdif  34418  hgt750lemd  34646  redvmptabs  42355