Theorem un0 2287

Description: The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27.

Assertion

Ref	Expression
un0	|- (A u. (/)) = A

Proof of Theorem un0

Step	Hyp	Ref	Expression
1		noel 2274	. . . 4 |- -. x e. (/)
2	1	biorfi 734	. . 3 |- (x e. A <-> (x e. A \/ x e. (/)))
3	2	bicomi 172	. 2 |- ((x e. A \/ x e. (/)) <-> x e. A)
4	3	uneqri 2164	1 |- (A u. (/)) = A

Syntax hints:   \/ wo 222   = wceq 953   e. wcel 955   u. cun 2035  (/)c0 2270

This theorem is referenced by:  un00 2296  difun2 2332  difdifdir 2336  prprc1 2443  unidif0 2729  suc0 3033  sucprc 3034  fvsnun1 3780  fvsnun2 3781  oev2 4146  oarec 4180  mapunen 4482  kmlem2 4738  kmlem3 4739  kmlem11 4747  cda0en 4897  dffsum 6936  dfisum 7127  acdc2lem2 7431  acdc5lem2 7434  ruclem6 7458  alephadd 7524  indistop 7590  indistps 7595  mapudiscn 10399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271

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