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Theorem posprs 16272
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 2471 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2471 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2ispos2 16271 . 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 467 1  |-  ( K  e.  Poset  ->  K  e.  Preset  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   A.wral 2756   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275    Preset cpreset 16249   Posetcpo 16263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-preset 16251  df-poset 16269
This theorem is referenced by:  posref  16274  isipodrs  16485  ordtrest2NEWlem  28802  ordtrest2NEW  28803  ordtconlem1  28804
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