Theorem dfin4 3826

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 3795	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		dfss4 3820	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵))
3	1, 2	mpbi 219	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)
4		difin 3823	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	4	difeq2i 3687	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	3, 5	eqtr3i 2634	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Syntax hints:   = wceq 1475   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554

This theorem is referenced by:  indif  3828  cnvin  5459  imain  5888  resin  6071  elcls  20687  cmmbl  23109  mbfeqalem  23215  itg1addlem4  23272  itg1addlem5  23273  inelsiga  29525  inelros  29563  topdifinffinlem  32371  poimirlem9  32588  mblfinlem4  32619  ismblfin  32620  cnambfre  32628  stoweidlem50  38943  saliincl  39221  sge0fodjrnlem  39309  meadjiunlem  39358  caragendifcl  39404