Theorem difdisj 2337

Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29.

Assertion

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

Proof of Theorem difdisj

Step	Hyp	Ref	Expression
1		inss1 2230	. 2 |- (A i^i B) (_ A
2		inssdif0 2333	. 2 |- ((A i^i B) (_ A <-> (A i^i (B \ A)) = (/))
3	1, 2	mpbi 189	1 |- (A i^i (B \ A)) = (/)

Syntax hints:   = wceq 956   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  undifv 2339  difdifdir 2346  fvsnun1 3795  fvsnun2 3796  undom 4438  sbthlem7 4453  sbthlem8 4454  fodomr 4483  mapdom2lem 4493  mapdom2 4494  limensuci 4506  phplem2 4509  pssnn 4534  unfi 4551  unfiOLD 4552  acdc2lem2 7489  acdc5lem2 7492  ruclem6 7515  infxpidmlem11 7562  grothprim 8783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  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 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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