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Theorem dfsdom2 6917
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 6799 . 2  |-  ~<  =  (  ~<_  \  ~~  )
2 sbthcl 6916 . . 3  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
32difeq2i 3233 . 2  |-  (  ~<_  \  ~~  )  =  (  ~<_  \  (  ~<_  i^i  `'  ~<_  ) )
4 difin 3348 . 2  |-  (  ~<_  \ 
(  ~<_  i^i  `'  ~<_  ) )  =  (  ~<_  \  `'  ~<_  )
51, 3, 43eqtri 2280 1  |-  ~<  =  (  ~<_  \  `'  ~<_  )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    \ cdif 3091    i^i cin 3093   `'ccnv 4625    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795
This theorem is referenced by:  brsdom2  6918
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799
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