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

Theorem un0 3458
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 3428 . . . 4  |-  -.  x  e.  (/)
21biorfi 746 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  \/  x  e.  (/) ) )
32bicomi 132 . 2  |-  ( ( x  e.  A  \/  x  e.  (/) )  <->  x  e.  A )
43uneqri 3279 1  |-  ( A  u.  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    \/ wo 708    = wceq 1353    e. wcel 2148    u. cun 3129   (/)c0 3424
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 2741  df-dif 3133  df-un 3135  df-nul 3425
This theorem is referenced by:  un00  3471  disjssun  3488  difun2  3504  difdifdirss  3509  disjpr2  3658  prprc1  3702  diftpsn3  3735  iununir  3972  exmid1stab  4210  suc0  4413  sucprc  4414  fvun1  5585  fmptpr  5711  fvunsng  5713  fvsnun1  5716  fvsnun2  5717  fsnunfv  5720  fsnunres  5721  rdg0  6391  omv2  6469  unsnfidcex  6922  unfidisj  6924  undifdc  6926  ssfirab  6936  dju0en  7216  djuassen  7219  fzsuc2  10082  fseq1p1m1  10097  hashunlem  10787  ennnfonelem1  12411  setsresg  12503  setsslid  12516  fmelpw1o  14719
  Copyright terms: Public domain W3C validator