Theorem inindif 30842

Description: See inundif 4417. (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 4168	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2	1	orci 861	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶)
3		inss 4177	. . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶)
4	2, 3	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶
5		inssdif0 4308	. 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
6	4, 5	mpbi 229	1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅

Syntax hints:   ∨ wo 843   = wceq 1541   ∖ cdif 3888   ∩ cin 3890   ⊆ wss 3891  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-in 3898  df-ss 3908  df-nul 4262

This theorem is referenced by:  resf1o  31044  gsummptres  31291  indsumin  31969  measunl  32163  carsgclctun  32267  probdif  32366  hgt750lemd  32607