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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 ∪ ∅) = 𝐴

Proof of Theorem un0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noel 3426 . . . 4 ¬ 𝑥 ∈ ∅
21biorfi 746 . . 3 (𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
32bicomi 132 . 2 ((𝑥𝐴𝑥 ∈ ∅) ↔ 𝑥𝐴)
43uneqri 3277 1 (𝐴 ∪ ∅) = 𝐴
Colors of variables: wff set class
Syntax hints:  wo 708   = wceq 1353  wcel 2148  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  3655  prprc1  3699  diftpsn3  3732  iununir  3967  exmid1stab  4205  suc0  4408  sucprc  4409  fvun1  5578  fmptpr  5704  fvunsng  5706  fvsnun1  5709  fvsnun2  5710  fsnunfv  5713  fsnunres  5714  rdg0  6382  omv2  6460  unsnfidcex  6913  unfidisj  6915  undifdc  6917  ssfirab  6927  dju0en  7207  djuassen  7210  fzsuc2  10062  fseq1p1m1  10077  hashunlem  10765  ennnfonelem1  12388  setsresg  12480  setsslid  12492  fmelpw1o  14207
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