Theorem disjdif 3476

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 3337	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		inssdif0im 3471	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1342   ∖ cdif 3108   ∩ cin 3110   ⊆ wss 3111  ∅c0 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-dif 3113  df-in 3117  df-ss 3124  df-nul 3405

This theorem is referenced by:  ssdifin0  3485  difdifdirss  3488  fvsnun1  5676  fvsnun2  5677  phplem2  6810  unfiin  6882  xpfi  6886  sbthlem7  6919  sbthlemi8  6920  exmidfodomrlemim  7148  fihashssdif  10720  zfz1isolem1  10739  fsumlessfi  11387  fprodsplit1f  11561  setsfun  12366  setsfun0  12367  setsslid  12381