Theorem difin0 2946

Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem difin0

Step	Hyp	Ref	Expression
1		inss2 2813	. 2 |- (A i^i B) C_ B
2		ssdif0 2934	. 2 |- ((A i^i B) C_ B <-> ((A i^i B) \ B) = (/))
3	1, 2	mpbi 206	1 |- ((A i^i B) \ B) = (/)

Syntax hints:   = wceq 1298   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875

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-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876

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