Theorem disjdif 3622

Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Assertion

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

Proof of Theorem disjdif

Step	Hyp	Ref	Expression
1		inss1 3475	. 2 |- (A i^i B) C_ A
2		inssdif0 3617	. 2 |- ((A i^i B) C_ A <-> (A i^i (B \ A)) = (/))
3	1, 2	mpbi 199	1 |- (A i^i (B \ A)) = (/)

Syntax hints:   = wceq 1642   \ cdif 3206   i^i cin 3208   C_ wss 3257  (/)c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  undifv  3624  ssdifin0  3631  difdifdir  3637  sfinltfin  4535  phialllem2  4617  fvsnun1  5447  fvsnun2  5448  enadj  6060  sbthlem1  6203  dflec2  6210