Theorem disjdif 3569

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 ∩ (𝐵 ∖ 𝐴)) = ∅

Proof of Theorem disjdif

Step	Hyp	Ref	Expression
1		inss1 3429	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		inssdif0im 3564	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1398   ∖ cdif 3198   ∩ cin 3200   ⊆ wss 3201  ∅c0 3496

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497

This theorem is referenced by:  ssdifin0  3578  difdifdirss  3581  fvsnun1  5859  fvsnun2  5860  phplem2  7082  unfiin  7161  xpfi  7167  sbthlem7  7205  sbthlemi8  7206  exmidfodomrlemim  7455  fihashssdif  11126  zfz1isolem1  11148  fsumlessfi  12082  fprodsplit1f  12256  setsfun  13178  setsfun0  13179  setsslid  13194