Theorem disjdif 3581

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 3441	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3576	. 2 |- ( ( A i^i B ) C_ A -> ( A i^i ( B \ A ) ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A i^i ( B \ A ) ) = (/)

Syntax hints:   = wceq 1398   \ cdif 3208   i^i cin 3210   C_ wss 3211   (/)c0 3508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509

This theorem is referenced by:  disjdifr  3582  ssdifin0  3591  difdifdirss  3594  fvsnun1  5881  fvsnun2  5882  phplem2  7107  unfiin  7186  xpfi  7192  sbthlem7  7233  sbthlemi8  7234  exmidfodomrlemim  7504  fihashssdif  11183  zfz1isolem1  11212  fsumlessfi  12146  fprodsplit1f  12320  setsfun  13247  setsfun0  13248  setsslid  13263