Theorem disjdif 3495

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 3355	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3490	. 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 1353   \ cdif 3126   i^i cin 3128   C_ wss 3129   (/)c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423

This theorem is referenced by:  ssdifin0  3504  difdifdirss  3507  fvsnun1  5713  fvsnun2  5714  phplem2  6852  unfiin  6924  xpfi  6928  sbthlem7  6961  sbthlemi8  6962  exmidfodomrlemim  7199  fihashssdif  10797  zfz1isolem1  10819  fsumlessfi  11467  fprodsplit1f  11641  setsfun  12496  setsfun0  12497  setsslid  12512