Theorem dfin4 4231

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 4190	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		dfss4 4222	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵))
3	1, 2	mpbi 230	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)
4		difin 4225	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	4	difeq2i 4076	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	3, 5	eqtr3i 2754	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Syntax hints:   = wceq 1540   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-in 3912  df-ss 3922

This theorem is referenced by:  indif  4233  cnvin  6097  imain  6571  resin  6790  elcls  22976  cmmbl  25451  mbfeqalem2  25559  itg1addlem4  25616  itg1addlem5  25617  suppovss  32637  inelsiga  34104  inelros  34142  topdifinffinlem  37323  poimirlem9  37611  mblfinlem4  37642  ismblfin  37643  cnambfre  37650  stoweidlem50  46035  saliinclf  46311  sge0fodjrnlem  46401  meadjiunlem  46450  caragendifcl  46499