Theorem difin0 2336

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 2258	. 2 |- ((A i^i B) \ B) = ((A \ B) i^i (B \ B))
2		difid 2332	. . 3 |- (B \ B) = (/)
3	2	ineq2i 2212	. 2 |- ((A \ B) i^i (B \ B)) = ((A \ B) i^i (/))
4		in0 2296	. 2 |- ((A \ B) i^i (/)) = (/)
5	1, 3, 4	3eqtr 1498	1 |- ((A i^i B) \ B) = (/)

Syntax hints:   = wceq 955   \ cdif 2042   i^i cin 2044  (/)c0 2278

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-in 2049  df-ss 2051  df-nul 2279

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