Theorem indif 4270

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 4268	. 2 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
2		dfin4 4268	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
3	2	difeq2i 4120	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵)))
4		difin 4262	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	1, 3, 4	3eqtr2i 2764	1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1539   ∖ cdif 3946   ∩ cin 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-in 3956  df-ss 3966

This theorem is referenced by:  resdif  6855  kmlem11  10159  psgndiflemB  21374