ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  un0 Unicode version

Theorem un0 3471
Description: The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
un0  |-  ( A  u.  (/) )  =  A

Proof of Theorem un0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3441 . . . 4  |-  -.  x  e.  (/)
21biorfi 747 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  \/  x  e.  (/) ) )
32bicomi 132 . 2  |-  ( ( x  e.  A  \/  x  e.  (/) )  <->  x  e.  A )
43uneqri 3292 1  |-  ( A  u.  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    \/ wo 709    = wceq 1364    e. wcel 2160    u. cun 3142   (/)c0 3437
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438
This theorem is referenced by:  un00  3484  disjssun  3501  difun2  3517  difdifdirss  3522  disjpr2  3671  prprc1  3715  diftpsn3  3748  iununir  3985  exmid1stab  4226  suc0  4429  sucprc  4430  fvun1  5602  fmptpr  5728  fvunsng  5730  fvsnun1  5733  fvsnun2  5734  fsnunfv  5737  fsnunres  5738  rdg0  6411  omv2  6489  unsnfidcex  6947  unfidisj  6949  undifdc  6951  ssfirab  6961  dju0en  7242  djuassen  7245  fzsuc2  10108  fseq1p1m1  10123  hashunlem  10815  ennnfonelem1  12457  setsresg  12549  setsslid  12562  fmelpw1o  15011
  Copyright terms: Public domain W3C validator