Theorem dfin4 3496

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 3476	. . 3 ⊢ (A ∩ B) ⊆ A
2		dfss4 3490	. . 3 ⊢ ((A ∩ B) ⊆ A ↔ (A ∖ (A ∖ (A ∩ B))) = (A ∩ B))
3	1, 2	mpbi 199	. 2 ⊢ (A ∖ (A ∖ (A ∩ B))) = (A ∩ B)
4		difin 3493	. . 3 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)
5	4	difeq2i 3383	. 2 ⊢ (A ∖ (A ∖ (A ∩ B))) = (A ∖ (A ∖ B))
6	3, 5	eqtr3i 2375	1 ⊢ (A ∩ B) = (A ∖ (A ∖ B))

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by:  indif  3498  cnvin  5036  imain  5173  resin  5308