Theorem disjdifb 47750

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 4266	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵)
2		disjdif 4466	. . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
3	2	difeq1i 4113	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4		0dif 4396	. 2 ⊢ (∅ ∖ 𝐵) = ∅
5	1, 3, 4	3eqtri 2758	1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1533   ∖ cdif 3940   ∩ cin 3942  ∅c0 4317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318

This theorem is referenced by:  iscnrm3r  47837