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Theorem dfsdom2 6979
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 6861 . 2  |-  ~<  =  (  ~<_  \  ~~  )
2 sbthcl 6978 . . 3  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
32difeq2i 3292 . 2  |-  (  ~<_  \  ~~  )  =  (  ~<_  \  (  ~<_  i^i  `'  ~<_  ) )
4 difin 3407 . 2  |-  (  ~<_  \ 
(  ~<_  i^i  `'  ~<_  ) )  =  (  ~<_  \  `'  ~<_  )
51, 3, 43eqtri 2308 1  |-  ~<  =  (  ~<_  \  `'  ~<_  )
Colors of variables: wff set class
Syntax hints:    = wceq 1624    \ cdif 3150    i^i cin 3152   `'ccnv 4687    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857
This theorem is referenced by:  brsdom2  6980
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861
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