Theorem dfin4 4278

Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)

Assertion

Ref	Expression
dfin4	⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Proof of Theorem dfin4

Step	Hyp	Ref	Expression
1		inss1 4237	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		dfss4 4269	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵))
3	1, 2	mpbi 230	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)
4		difin 4272	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	4	difeq2i 4123	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	3, 5	eqtr3i 2767	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Syntax hints:   = wceq 1540   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968

This theorem is referenced by:  indif  4280  cnvin  6164  imain  6651  resin  6870  elcls  23081  cmmbl  25569  mbfeqalem2  25677  itg1addlem4  25734  itg1addlem5  25735  suppovss  32690  inelsiga  34136  inelros  34174  topdifinffinlem  37348  poimirlem9  37636  mblfinlem4  37667  ismblfin  37668  cnambfre  37675  stoweidlem50  46065  saliinclf  46341  sge0fodjrnlem  46431  meadjiunlem  46480  caragendifcl  46529