MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdomdom Structured version   Visualization version   GIF version

Theorem sdomdom 7928
Description: Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
Assertion
Ref Expression
sdomdom (𝐴𝐵𝐴𝐵)

Proof of Theorem sdomdom
StepHypRef Expression
1 brsdom 7923 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
21simplbi 476 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   class class class wbr 4618  cen 7897  cdom 7898  csdm 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-br 4619  df-sdom 7903
This theorem is referenced by:  domdifsn  7988  sdomnsym  8030  sdomdomtr  8038  domsdomtr  8040  sdomtr  8043  sucdom2  8101  sucxpdom  8114  isfinite2  8163  pwfi  8206  card2on  8404  fict  8495  fidomtri2  8765  prdom2  8774  infxpenlem  8781  indcardi  8809  alephnbtwn2  8840  alephsucdom  8847  alephdom  8849  dfac13  8909  cdalepw  8963  infcdaabs  8973  infdif  8976  infunsdom1  8980  infunsdom  8981  infxp  8982  cfslb2n  9035  sdom2en01  9069  isfin32i  9132  fin34  9157  fin67  9162  hsmexlem1  9193  hsmex3  9201  entri3  9326  unirnfdomd  9334  alephexp1  9346  cfpwsdom  9351  gchdomtri  9396  canthp1  9421  pwfseqlem5  9430  gchcdaidm  9435  gchxpidm  9436  gchpwdom  9437  hargch  9440  gchaclem  9445  gchhar  9446  gchac  9448  inawinalem  9456  inar1  9542  rankcf  9544  tskuni  9550  grothac  9597  rpnnen  14876  rexpen  14877  aleph1irr  14895  dis1stc  21207  hauspwdom  21209  ovolfi  23164  sibfof  30175  heiborlem3  33230  harinf  37067  saluncl  39831  meadjun  39973  meaiunlelem  39979  omeunle  40024
  Copyright terms: Public domain W3C validator