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Theorem sdomnsym 6982
Description: Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
sdomnsym  |-  ( A 
~<  B  ->  -.  B  ~<  A )

Proof of Theorem sdomnsym
StepHypRef Expression
1 sdomnen 6886 . 2  |-  ( A 
~<  B  ->  -.  A  ~~  B )
2 sdomdom 6885 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
3 sdomdom 6885 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
4 sbth 6977 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
52, 3, 4syl2an 465 . 2  |-  ( ( A  ~<  B  /\  B  ~<  A )  ->  A  ~~  B )
61, 5mtand 642 1  |-  ( A 
~<  B  ->  -.  B  ~<  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   class class class wbr 4025    ~~ cen 6856    ~<_ cdom 6857    ~< csdm 6858
This theorem is referenced by:  domnsym  6983  gchpwdom  8292
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 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-en 6860  df-dom 6861  df-sdom 6862
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