Theorem un0 3251

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 3228	. . . 4 |- -. x e. (/)
2	1	biorfi 665	. . 3 |- ( x e. A <-> ( x e. A \/ x e. (/) ) )
3	2	bicomi 123	. 2 |- ( ( x e. A \/ x e. (/) ) <-> x e. A )
4	3	uneqri 3085	1 |- ( A u. (/) ) = A

Syntax hints:   \/ wo 629   = wceq 1243   e. wcel 1393   u. cun 2915   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225

This theorem is referenced by:  un00  3263  disjssun  3285  difun2  3302  difdifdirss  3307  disjpr2  3434  prprc1  3478  diftpsn3  3505  iununir  3738  suc0  4148  sucprc  4149  fvun1  5239  fmptpr  5355  fvunsng  5357  fvsnun1  5360  fvsnun2  5361  fsnunfv  5363  fsnunres  5364  rdg0  5974  omv2  6045  fzsuc2  8941  fseq1p1m1  8956