Theorem dfin4 2835

Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231.

Assertion

Ref	Expression
dfin4	|- (A i^i B) = (A \ (A \ B))

Proof of Theorem dfin4

Step	Hyp	Ref	Expression
1		inss1 2812	. . 3 |- (A i^i B) C_ A
2		dfss4 2827	. . 3 |- ((A i^i B) C_ A <-> (A \ (A \ (A i^i B))) = (A i^i B))
3	1, 2	mpbi 206	. 2 |- (A \ (A \ (A i^i B))) = (A i^i B)
4		difin 2831	. . 3 |- (A \ (A i^i B)) = (A \ B)
5	4	difeq2i 2723	. 2 |- (A \ (A \ (A i^i B))) = (A \ (A \ B))
6	3, 5	eqtr3i 1910	1 |- (A i^i B) = (A \ (A \ B))

Syntax hints:   = wceq 1298   \ cdif 2590   i^i cin 2592   C_ wss 2593

This theorem is referenced by:  indif 2837  imain 4494  resin 4660  elcls 8980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605

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