Theorem un0 3285

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.

Step	Hyp	Ref	Expression
1		noel 3262	. . . 4 |- -. x e. (/)
2	1	biorfi 698	. . 3 |- ( x e. A <-> ( x e. A \/ x e. (/) ) )
3	2	bicomi 130	. 2 |- ( ( x e. A \/ x e. (/) ) <-> x e. A )
4	3	uneqri 3115	1 |- ( A u. (/) ) = A

Syntax hints:   \/ wo 662   = wceq 1285   e. wcel 1434   u. cun 2972   (/)c0 3258

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259

This theorem is referenced by:  un00  3297  disjssun  3314  difun2  3329  difdifdirss  3334  disjpr2  3464  prprc1  3508  diftpsn3  3535  iununir  3767  suc0  4174  sucprc  4175  fvun1  5271  fmptpr  5387  fvunsng  5389  fvsnun1  5392  fvsnun2  5393  fsnunfv  5395  fsnunres  5396  rdg0  6036  omv2  6109  unsnfidcex  6440  unfidisj  6442  undiffi  6443  fzsuc2  9172  fseq1p1m1  9187  sizeunlem  9828