Theorem indif 4279

Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
indif	⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Proof of Theorem indif

Step	Hyp	Ref	Expression
1		dfin4 4277	. 2 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
2		dfin4 4277	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
3	2	difeq2i 4122	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
4		difin 4271	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	1, 3, 4	3eqtr2i 2770	1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1539   ∖ cdif 3947   ∩ cin 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967

This theorem is referenced by:  resdif  6868  kmlem11  10202  psgndiflemB  21619