MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abid2 Unicode version

Theorem abid2 2366
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
abid2  |-  { x  |  x  e.  A }  =  A
Distinct variable group:    x, A

Proof of Theorem abid2
StepHypRef Expression
1 biid 229 . . 3  |-  ( x  e.  A  <->  x  e.  A )
21abbi2i 2360 . 2  |-  A  =  { x  |  x  e.  A }
32eqcomi 2257 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   {cab 2239
This theorem is referenced by:  csbid  3016  csbexg  3019  abss  3163  ssab  3164  abssi  3169  notab  3345  inrab2  3348  dfrab2  3350  dfrab3  3351  notrab  3352  eusn  3607  uniintsn  3797  iunid  3855  orduniss2  4515  imai  4934  dffv3  6171  riotav  6195  cbvriota  6201  riotaund  6227  dfixp  6705  euen1b  6817  modom2  6949  infmap2  7728  dffv4  23637  mapex2  24306  prismorcsetlemb  25079  aomclem4  26320  rngunsnply  26544  compneOLD  26810  pmapglb  28863  polval2N  28999
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249
  Copyright terms: Public domain W3C validator