Theorem un0 2297

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 2284	. . . 4 |- -. x e. (/)
2	1	biorfi 736	. . 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 2174	1 |- (A u. (/)) = A

Syntax hints:   \/ wo 222   = wceq 956   e. wcel 958   u. cun 2045  (/)c0 2280

This theorem is referenced by:  un00 2306  difun2 2342  difdifdir 2346  prprc1 2452  unidif0 2739  suc0 3043  sucprc 3044  fvsnun1 3795  fvsnun2 3796  oev2 4162  oarec 4196  mapunen 4502  kmlem2 4766  kmlem3 4767  kmlem11 4775  cda0en 4925  dffsum 6998  dfisum 7191  acdc2lem2 7489  acdc5lem2 7492  ruclem6 7515  alephadd 7582  indistop 7648  indistps 7653  mapudiscn 10512  eqindhome 10541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281

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