Theorem indif 4196

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 4194	. 2 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
2		dfin4 4194	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
3	2	difeq2i 4047	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
4		difin 4188	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	1, 3, 4	3eqtr2i 2827	1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1538   ∖ cdif 3878   ∩ cin 3880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898

This theorem is referenced by:  resdif  6610  kmlem11  9571  psgndiflemB  20289