Theorem disjdif 3580

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 3440	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		inssdif0im 3575	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1398   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3210  ∅c0 3507

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 2814  df-dif 3212  df-in 3216  df-ss 3223  df-nul 3508

This theorem is referenced by:  disjdifr  3581  ssdifin0  3590  difdifdirss  3593  fvsnun1  5880  fvsnun2  5881  phplem2  7106  unfiin  7185  xpfi  7191  sbthlem7  7232  sbthlemi8  7233  exmidfodomrlemim  7503  fihashssdif  11181  zfz1isolem1  11208  fsumlessfi  12142  fprodsplit1f  12316  setsfun  13239  setsfun0  13240  setsslid  13255