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

Theorem reldom 7115
Description: Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
reldom  |-  Rel  ~<_

Proof of Theorem reldom
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dom 7111 . 2  |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
21relopabi 5000 1  |-  Rel  ~<_
Colors of variables: wff set class
Syntax hints:   E.wex 1550   Rel wrel 4883   -1-1->wf1 5451    ~<_ cdom 7107
This theorem is referenced by:  relsdom  7116  brdomg  7118  brdomi  7119  domtr  7160  undom  7196  xpdom2  7203  xpdom1g  7205  domunsncan  7208  sbth  7227  sbthcl  7229  dom0  7235  fodomr  7258  pwdom  7259  domssex  7268  mapdom1  7272  mapdom2  7278  fineqv  7324  infsdomnn  7368  infn0  7369  elharval  7531  harword  7533  domwdom  7542  unxpwdom  7557  infdifsn  7611  infdiffi  7612  ac10ct  7915  iunfictbso  7995  cdadom1  8066  cdainf  8072  infcda1  8073  pwcdaidm  8075  cdalepw  8076  unctb  8085  infcdaabs  8086  infunabs  8087  infpss  8097  infmap2  8098  fictb  8125  infpssALT  8193  fin34  8270  ttukeylem1  8389  fodomb  8404  wdomac  8405  brdom3  8406  iundom2g  8415  iundom  8417  infxpidm  8437  iunctb  8449  gchdomtri  8504  pwfseq  8539  pwxpndom2  8540  pwxpndom  8541  pwcdandom  8542  gchaclem  8545  gchpwdom  8549  reexALT  10606  hashdomi  11654  cctop  17070  1stcrestlem  17515  2ndcdisj2  17520  dis2ndc  17523  hauspwdom  17564  ufilen  17962  ovoliunnul  19403  uniiccdif  19470  ovoliunnfl  26248  voliunnfl  26250  volsupnfl  26251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885  df-dom 7111
  Copyright terms: Public domain W3C validator