Theorem un0 3245

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 3222	. . . 4 |- -. x e. (/)
2	1	biorfi 664	. . 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 3079	1 |- (A u. (/)) = A

Syntax hints:   \/ wo 628   = wceq 1242   e. wcel 1390   u. cun 2909   (/)c0 3218

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219

This theorem is referenced by:  un00  3257  disjssun  3279  difun2  3296  difdifdirss  3301  disjpr2  3425  prprc1  3469  diftpsn3  3496  iununir  3729  suc0  4114  sucprc  4115  fvun1  5182  fmptpr  5298  fvunsng  5300  fvsnun1  5303  fvsnun2  5304  fsnunfv  5306  fsnunres  5307  rdg0  5914  omv2  5984  fzsuc2  8711  fseq1p1m1  8726