Is Real Numbers an Ordered Field Under <?

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Sir,i have read in wikipedia that for a relation to be ordered it should be transitive,antisymmetric,total...however we know that Real numbers is an ordered field under relation "<" but antisymmetric property is not valid with "<" relation..how is this justified..rectify me...
 
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The real numbers are not an ordered field under <.
Rather, the real numbers are an ordered field under \leq.
 

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