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Theorem posprs 14083
Description: A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
posprs  |-  ( K  e.  Poset  ->  K  e.  Preset  )

Proof of Theorem posprs
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2ispos2 14082 . 2  |-  ( K  e.  Poset 
<->  ( K  e.  Preset  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) ( ( x ( le
`  K ) y  /\  y ( le
`  K ) x )  ->  x  =  y ) ) )
43simplbi 446 1  |-  ( K  e.  Poset  ->  K  e.  Preset  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215    Preset cpreset 14060   Posetcpo 14074
This theorem is referenced by:  isipodrs  14264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-preset 14062  df-poset 14080
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