Theorem difid 2338

Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.

Assertion

Ref	Expression
difid	|- (A \ A) = (/)

Proof of Theorem difid

Step	Hyp	Ref	Expression
1		ssid 2083	. 2 |- A (_ A
2		ssdif0 2331	. 2 |- (A (_ A <-> (A \ A) = (/))
3	1, 2	mpbi 189	1 |- (A \ A) = (/)

Syntax hints:   = wceq 958   \ cdif 2047   (_ wss 2050  (/)c0 2283

This theorem is referenced by:  dif0 2339  difin0 2342  difun2 2346  oev2 4168  topcld 7672

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284

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