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Theorem un0 3428
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 3399 . . . 4  |-  -.  x  e.  (/)
21biorfi 736 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  \/  x  e.  (/) ) )
32bicomi 131 . 2  |-  ( ( x  e.  A  \/  x  e.  (/) )  <->  x  e.  A )
43uneqri 3250 1  |-  ( A  u.  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    \/ wo 698    = wceq 1335    e. wcel 2128    u. cun 3100   (/)c0 3395
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-nul 3396
This theorem is referenced by:  un00  3441  disjssun  3458  difun2  3474  difdifdirss  3479  disjpr2  3625  prprc1  3669  diftpsn3  3699  iununir  3934  suc0  4373  sucprc  4374  fvun1  5536  fmptpr  5661  fvunsng  5663  fvsnun1  5666  fvsnun2  5667  fsnunfv  5670  fsnunres  5671  rdg0  6336  omv2  6414  unsnfidcex  6866  unfidisj  6868  undifdc  6870  ssfirab  6880  dju0en  7151  djuassen  7154  fzsuc2  9987  fseq1p1m1  10002  hashunlem  10689  ennnfonelem1  12206  setsresg  12298  setsslid  12310  fmelpw1o  13452  exmid1stab  13643
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