Theorem inindif 31742

Description: See inundif 4478. (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 4229	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2	1	orci 864	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶)
3		inss 4238	. . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶)
4	2, 3	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶
5		inssdif0 4369	. 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
6	4, 5	mpbi 229	1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅

Syntax hints:   ∨ wo 846   = wceq 1542   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  resf1o  31943  gsummptres  32192  indsumin  33009  measunl  33203  carsgclctun  33309  probdif  33408  hgt750lemd  33649