Theorem disjdifb 49300

Description: Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)

Assertion

Ref	Expression
disjdifb	⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅

Proof of Theorem disjdifb

Step	Hyp	Ref	Expression
1		indif1 4210	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵)
2		disjdif 4400	. . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
3	2	difeq1i 4053	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4		0dif 4333	. 2 ⊢ (∅ ∖ 𝐵) = ∅
5	1, 3, 4	3eqtri 2766	1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1547   ∖ cdif 3880   ∩ cin 3882  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262

This theorem is referenced by:  iscnrm3r  49438