Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldm2 Unicode version

Theorem eldm2 5206
 Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
Hypothesis
Ref Expression
eldm.1
Assertion
Ref Expression
eldm2
Distinct variable groups:   ,   ,

Proof of Theorem eldm2
StepHypRef Expression
1 eldm.1 . 2
2 eldm2g 5204 . 2
31, 2ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  E.wex 1612  e.wcel 1818   cvv 3109  <.cop 4035  domcdm 5004 This theorem is referenced by:  dmss  5207  opeldm  5211  dmin  5215  dmiun  5216  dmuni  5217  dm0  5221  reldm0  5225  dmrnssfld  5266  dmcoss  5267  dmcosseq  5269  dmres  5299  iss  5326  dmsnopg  5484  relssdmrn  5533  funssres  5633  dmfco  5947  fun11iun  6760  axdc3lem2  8852  gsum2d2  17002  cnlnssadj  26999  prsdm  27896  eldm3  29191  dfdm5  29206  wfrlem12  29354  frrlem11  29399  tfrqfree  29601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-dm 5014
 Copyright terms: Public domain W3C validator