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Theorem un0 3456
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 3426 . . . 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 3277 1  |-  ( A  u.  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    \/ wo 708    = wceq 1353    e. wcel 2148    u. 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-nul 3423
This theorem is referenced by:  un00  3469  disjssun  3486  difun2  3502  difdifdirss  3507  disjpr2  3656  prprc1  3700  diftpsn3  3733  iununir  3970  exmid1stab  4208  suc0  4411  sucprc  4412  fvun1  5582  fmptpr  5708  fvunsng  5710  fvsnun1  5713  fvsnun2  5714  fsnunfv  5717  fsnunres  5718  rdg0  6387  omv2  6465  unsnfidcex  6918  unfidisj  6920  undifdc  6922  ssfirab  6932  dju0en  7212  djuassen  7215  fzsuc2  10078  fseq1p1m1  10093  hashunlem  10783  ennnfonelem1  12407  setsresg  12499  setsslid  12512  fmelpw1o  14528
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