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Theorem dfsdom2 7637
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 7516 . 2  |-  ~<  =  (  ~<_  \  ~~  )
2 sbthcl 7636 . . 3  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
32difeq2i 3619 . 2  |-  (  ~<_  \  ~~  )  =  (  ~<_  \  (  ~<_  i^i  `'  ~<_  ) )
4 difin 3735 . 2  |-  (  ~<_  \ 
(  ~<_  i^i  `'  ~<_  ) )  =  (  ~<_  \  `'  ~<_  )
51, 3, 43eqtri 2500 1  |-  ~<  =  (  ~<_  \  `'  ~<_  )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    i^i cin 3475   `'ccnv 4998    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by:  brsdom2  7638
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