Theorem abid2 1186

Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.

Assertion

Ref	Expression
abid2	|- {x | x e. A} = A

Distinct variable group(s):   x,A

Proof of Theorem abid2

Step	Hyp	Ref	Expression
1		pm4.2 148	. . 3 |- (x e. A <-> x e. A)
2	1	biabri 1180	. 2 |- A = {x | x e. A}
3	2	cleqcomi 1105	1 |- {x | x e. A} = A

Syntax hints:  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  ssab 1555  dfrab2 1696  opabss 2100  dfepfr 2184  epfrc 2185  dmexg 2551  rnexg 2569  imai 2613  ecid 3236  qsid 3237  cardval 3633  cardval2 3661  cfsuc 3709  nnind 4434  infmap2 4953

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org