Theorem disjdif 3534

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 3394	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		inssdif0im 3529	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1373   ∖ cdif 3164   ∩ cin 3166   ⊆ wss 3167  ∅c0 3461

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3169  df-in 3173  df-ss 3180  df-nul 3462

This theorem is referenced by:  ssdifin0  3543  difdifdirss  3546  fvsnun1  5788  fvsnun2  5789  phplem2  6957  unfiin  7030  xpfi  7036  sbthlem7  7072  sbthlemi8  7073  exmidfodomrlemim  7316  fihashssdif  10970  zfz1isolem1  10992  fsumlessfi  11815  fprodsplit1f  11989  setsfun  12911  setsfun0  12912  setsslid  12927