Theorem un0 1721

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 1711	. . . 4 |- -. x e. (/)
2	1	biorfi 552	. . 3 |- (x e. A <-> (x e. A \/ x e. (/)))
3	2	bicomi 150	. 2 |- ((x e. A \/ x e. (/)) <-> x e. A)
4	3	uneqri 1602	1 |- (A u. (/)) = A

Syntax hints:   \/ wo 195   = wceq 1091   e. wcel 1092   u. cun 1485  (/)c0 1707

This theorem is referenced by:  un00 1728  difun2 1763  difdifdir 1765  prprc 1839  unidif0 1944  sucprc 2297  df1o2 3111  mapunen 3397  kmlem2 3581  kmlem3 3582  kmlem10 3589  cda0en 3720  facnnt 4870  fac0 4871  ruclem6 4890  ruclem7 4891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708

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