Theorem un0 3528

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 3498	. . . 4 |- -. x e. (/)
2	1	biorfi 753	. . 3 |- ( x e. A <-> ( x e. A \/ x e. (/) ) )
3	2	bicomi 132	. 2 |- ( ( x e. A \/ x e. (/) ) <-> x e. A )
4	3	uneqri 3349	1 |- ( A u. (/) ) = A

Syntax hints:   \/ wo 715   = wceq 1397   e. wcel 2202   u. cun 3198   (/)c0 3494

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495

This theorem is referenced by:  un00  3541  disjssun  3558  difun2  3574  difdifdirss  3579  disjpr2  3733  prprc1  3780  diftpsn3  3814  iununir  4054  exmid1stab  4298  suc0  4508  sucprc  4509  fvun1  5712  fmptpr  5845  fvunsng  5847  fvsnun1  5850  fvsnun2  5851  fsnunfv  5854  fsnunres  5855  rdg0  6552  omv2  6632  unsnfidcex  7111  unfidisj  7113  undifdc  7115  ssfirab  7128  dju0en  7428  djuassen  7431  fmelpw1o  7464  fzsuc2  10313  fseq1p1m1  10328  hashunlem  11066  ennnfonelem1  13027  setsresg  13119  setsslid  13132  lgsquadlem2  15806