ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difun2 GIF version

Theorem difun2 3502
Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
difun2 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Proof of Theorem difun2
StepHypRef Expression
1 difundir 3388 . 2 ((𝐴𝐵) ∖ 𝐵) = ((𝐴𝐵) ∪ (𝐵𝐵))
2 difid 3491 . . 3 (𝐵𝐵) = ∅
32uneq2i 3286 . 2 ((𝐴𝐵) ∪ (𝐵𝐵)) = ((𝐴𝐵) ∪ ∅)
4 un0 3456 . 2 ((𝐴𝐵) ∪ ∅) = (𝐴𝐵)
51, 3, 43eqtri 2202 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cdif 3126  cun 3127  c0 3422
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  uneqdifeqim  3508  difprsn1  3731  orddif  4545  fisseneq  6928  dfn2  9185
  Copyright terms: Public domain W3C validator