Theorem disjdifb 48934

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 4231	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵)
2		disjdif 4421	. . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
3	2	difeq1i 4071	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4		0dif 4354	. 2 ⊢ (∅ ∖ 𝐵) = ∅
5	1, 3, 4	3eqtri 2760	1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3895   ∩ cin 3897  ∅c0 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4283

This theorem is referenced by:  iscnrm3r  49072