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

Theorem sdomirr 8367
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 8252 . . 3 (𝐴𝐴 → ¬ 𝐴𝐴)
2 enrefg 8255 . . 3 (𝐴 ∈ V → 𝐴𝐴)
31, 2nsyl3 136 . 2 (𝐴 ∈ V → ¬ 𝐴𝐴)
4 relsdom 8230 . . . 4 Rel ≺
54brrelex1i 5394 . . 3 (𝐴𝐴𝐴 ∈ V)
65con3i 152 . 2 𝐴 ∈ V → ¬ 𝐴𝐴)
73, 6pm2.61i 177 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2166  Vcvv 3415   class class class wbr 4874  cen 8220  csdm 8222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-en 8224  df-dom 8225  df-sdom 8226
This theorem is referenced by:  sdomn2lp  8369  2pwuninel  8385  2pwne  8386  r111  8916  alephval2  9710  alephom  9723  csdfil  22069
  Copyright terms: Public domain W3C validator