Theorem disjdif 3599

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		ssid 3268	. 2 ⊢ 𝐴 ⊆ 𝐴
2		disjdifg 3598	. 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅

Syntax hints:   = wceq 1402   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  disjdifr  3600  ssdifin0  3609  difdifdirss  3612  fvsnun1  5906  fvsnun2  5907  phplem2  7148  unfiin  7227  xpfi  7233  sbthlem7  7274  sbthlemi8  7275  exmidfodomrlemim  7547  fihashssdif  11242  hashf1lem2  11269  zfz1isolem1  11275  fsumlessfi  12210  fprodsplit1f  12384  setsfun  13370  setsfun0  13371  setsslid  13386