Theorem difin0 1759

Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29.

Assertion

Ref	Expression
difin0	|- ((A i^i B) \ B) = (/)

Proof of Theorem difin0

Step	Hyp	Ref	Expression
1		difindir 1684	. 2 |- ((A i^i B) \ B) = ((A \ B) i^i (B \ B))
2		difid 1755	. . 3 |- (B \ B) = (/)
3	2	ineq2i 1642	. 2 |- ((A \ B) i^i (B \ B)) = ((A \ B) i^i (/))
4		in0 1722	. 2 |- ((A \ B) i^i (/)) = (/)
5	1, 3, 4	3eqtr 1123	1 |- ((A i^i B) \ B) = (/)

Syntax hints:   = wceq 1091   \ cdif 1484   i^i cin 1486  (/)c0 1707

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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