Theorem dfin4 4239

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 4197	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		dfss4 4230	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵))
3	1, 2	mpbi 233	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)
4		difin 4233	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	4	difeq2i 4086	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	3, 5	eqtr3i 2794	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930

This theorem is referenced by:  indif  4241  cnvin  6139  imain  6618  resin  6841  elcls  23195  cmmbl  25658  mbfeqalem2  25766  itg1addlem4  25823  itg1addlem5  25824  suppovss  32963  inelsiga  34466  inelros  34504  topdifinffinlem  37876  poimirlem9  38163  mblfinlem4  38194  ismblfin  38195  cnambfre  38202  stoweidlem50  46649  saliinclf  46925  sge0fodjrnlem  47015  meadjiunlem  47064  caragendifcl  47113