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Theorem dfsdom2 7168
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 7050 . 2  |-  ~<  =  (  ~<_  \  ~~  )
2 sbthcl 7167 . . 3  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
32difeq2i 3407 . 2  |-  (  ~<_  \  ~~  )  =  (  ~<_  \  (  ~<_  i^i  `'  ~<_  ) )
4 difin 3523 . 2  |-  (  ~<_  \ 
(  ~<_  i^i  `'  ~<_  ) )  =  (  ~<_  \  `'  ~<_  )
51, 3, 43eqtri 2413 1  |-  ~<  =  (  ~<_  \  `'  ~<_  )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3262    i^i cin 3264   `'ccnv 4819    ~~ cen 7044    ~<_ cdom 7045    ~< csdm 7046
This theorem is referenced by:  brsdom2  7169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050
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