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

Theorem sdomirr 7982
Description: Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
Assertion
Ref Expression
sdomirr ¬ 𝐴𝐴

Proof of Theorem sdomirr
StepHypRef Expression
1 sdomnen 7870 . . 3 (𝐴𝐴 → ¬ 𝐴𝐴)
2 enrefg 7873 . . 3 (𝐴 ∈ V → 𝐴𝐴)
31, 2nsyl3 132 . 2 (𝐴 ∈ V → ¬ 𝐴𝐴)
4 relsdom 7848 . . . 4 Rel ≺
54brrelexi 5082 . . 3 (𝐴𝐴𝐴 ∈ V)
65con3i 149 . 2 𝐴 ∈ V → ¬ 𝐴𝐴)
73, 6pm2.61i 175 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1977  Vcvv 3173   class class class wbr 4583  cen 7838  csdm 7840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-en 7842  df-dom 7843  df-sdom 7844
This theorem is referenced by:  sdomn2lp  7984  2pwuninel  8000  2pwne  8001  r111  8521  alephval2  9273  alephom  9286  csdfil  21508
  Copyright terms: Public domain W3C validator