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Theorem sdomirr 7647
Description: Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
Assertion
Ref Expression
sdomirr  |-  -.  A  ~<  A

Proof of Theorem sdomirr
StepHypRef Expression
1 sdomnen 7537 . . 3  |-  ( A 
~<  A  ->  -.  A  ~~  A )
2 enrefg 7540 . . 3  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2nsyl3 119 . 2  |-  ( A  e.  _V  ->  -.  A  ~<  A )
4 relsdom 7516 . . . 4  |-  Rel  ~<
54brrelexi 5029 . . 3  |-  ( A 
~<  A  ->  A  e. 
_V )
65con3i 135 . 2  |-  ( -.  A  e.  _V  ->  -.  A  ~<  A )
73, 6pm2.61i 164 1  |-  -.  A  ~<  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1823   _Vcvv 3106   class class class wbr 4439    ~~ cen 7506    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by:  sdomn2lp  7649  2pwuninel  7665  2pwne  7666  r111  8184  alephval2  8938  alephom  8951  csdfil  20564
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