Theorem dfin4 4233

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 4191	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		dfss4 4224	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵))
3	1, 2	mpbi 233	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)
4		difin 4227	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
5	4	difeq2i 4080	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	3, 5	eqtr3i 2790	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))

Syntax hints:   = wceq 1563   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-in 3914  df-ss 3924

This theorem is referenced by:  indif  4235  cnvin  6132  imain  6610  resin  6833  elcls  23191  cmmbl  25654  mbfeqalem2  25762  itg1addlem4  25819  itg1addlem5  25820  suppovss  32938  inelsiga  34442  inelros  34480  topdifinffinlem  37853  poimirlem9  38140  mblfinlem4  38171  ismblfin  38172  cnambfre  38179  stoweidlem50  46622  saliinclf  46898  sge0fodjrnlem  46988  meadjiunlem  47037  caragendifcl  47086