Theorem un0 3525

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 3495	. . . 4 |- -. x e. (/)
2	1	biorfi 751	. . 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 3346	1 |- ( A u. (/) ) = A

Syntax hints:   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3195   (/)c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492

This theorem is referenced by:  un00  3538  disjssun  3555  difun2  3571  difdifdirss  3576  disjpr2  3730  prprc1  3774  diftpsn3  3808  iununir  4048  exmid1stab  4291  suc0  4501  sucprc  4502  fvun1  5699  fmptpr  5830  fvunsng  5832  fvsnun1  5835  fvsnun2  5836  fsnunfv  5839  fsnunres  5840  rdg0  6531  omv2  6609  unsnfidcex  7078  unfidisj  7080  undifdc  7082  ssfirab  7094  dju0en  7392  djuassen  7395  fmelpw1o  7428  fzsuc2  10271  fseq1p1m1  10286  hashunlem  11021  ennnfonelem1  12973  setsresg  13065  setsslid  13078  lgsquadlem2  15751