Theorem disjdif 3401

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 3262	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3396	. 2 |- ( ( A i^i B ) C_ A -> ( A i^i ( B \ A ) ) = (/) )
3	1, 2	ax-mp 7	1 |- ( A i^i ( B \ A ) ) = (/)

Syntax hints:   = wceq 1314   \ cdif 3034   i^i cin 3036   C_ wss 3037   (/)c0 3329

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-nul 3330

This theorem is referenced by:  ssdifin0  3410  difdifdirss  3413  fvsnun1  5571  fvsnun2  5572  phplem2  6700  unfiin  6767  xpfi  6771  sbthlem7  6803  sbthlemi8  6804  exmidfodomrlemim  7005  fihashssdif  10454  zfz1isolem1  10473  fsumlessfi  11118  setsfun  11834  setsfun0  11835  setsslid  11849