Theorem dfin4 2244

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 2226	. . 3 |- (A i^i B) (_ A
2		dfss4 2238	. . 3 |- ((A i^i B) (_ A <-> (A \ (A \ (A i^i B))) = (A i^i B))
3	1, 2	mpbi 189	. 2 |- (A \ (A \ (A i^i B))) = (A i^i B)
4		difin 2241	. . 3 |- (A \ (A i^i B)) = (A \ B)
5	4	difeq2i 2152	. 2 |- (A \ (A \ (A i^i B))) = (A \ (A \ B))
6	3, 5	eqtr3 1494	1 |- (A i^i B) = (A \ (A \ B))

Syntax hints:   = wceq 954   \ cdif 2040   i^i cin 2042   (_ wss 2043

This theorem is referenced by:  indif 2246  elcls 7654

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049

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