Theorem disjdifb 49608

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 4235	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵)
2		disjdif 4433	. . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
3	2	difeq1i 4077	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4		0dif 4363	. 2 ⊢ (∅ ∖ 𝐵) = ∅
5	1, 3, 4	3eqtri 2790	1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1570   ∖ cdif 3902   ∩ cin 3904  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287

This theorem is referenced by:  iscnrm3r  49746