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Theorem dfsdom2 7633
Description: Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
dfsdom2  |-  ~<  =  (  ~<_  \  `'  ~<_  )

Proof of Theorem dfsdom2
StepHypRef Expression
1 df-sdom 7512 . 2  |-  ~<  =  (  ~<_  \  ~~  )
2 sbthcl 7632 . . 3  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
32difeq2i 3605 . 2  |-  (  ~<_  \  ~~  )  =  (  ~<_  \  (  ~<_  i^i  `'  ~<_  ) )
4 difin 3732 . 2  |-  (  ~<_  \ 
(  ~<_  i^i  `'  ~<_  ) )  =  (  ~<_  \  `'  ~<_  )
51, 3, 43eqtri 2487 1  |-  ~<  =  (  ~<_  \  `'  ~<_  )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    \ cdif 3458    i^i cin 3460   `'ccnv 4987    ~~ cen 7506    ~<_ cdom 7507    ~< 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-er 7303  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by:  brsdom2  7634
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