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Theorem sdomnsym 7218
Description: Strict dominance is asymmetric. 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 7122 . 2  |-  ( A 
~<  B  ->  -.  A  ~~  B )
2 sdomdom 7121 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
3 sdomdom 7121 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
4 sbth 7213 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
52, 3, 4syl2an 464 . 2  |-  ( ( A  ~<  B  /\  B  ~<  A )  ->  A  ~~  B )
61, 5mtand 641 1  |-  ( A 
~<  B  ->  -.  B  ~<  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   class class class wbr 4199    ~~ cen 7092    ~<_ cdom 7093    ~< csdm 7094
This theorem is referenced by:  domnsym  7219  gchpwdom  8533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-en 7096  df-dom 7097  df-sdom 7098
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