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Theorem isposi 14444
Description: Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
isposi.k  |-  K  e. 
_V
isposi.b  |-  B  =  ( Base `  K
)
isposi.l  |-  .<_  =  ( le `  K )
isposi.1  |-  ( x  e.  B  ->  x  .<_  x )
isposi.2  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y ) )
isposi.3  |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )
Assertion
Ref Expression
isposi  |-  K  e. 
Poset
Distinct variable groups:    x, y,
z, B    x,  .<_ , y, z
Allowed substitution hints:    K( x, y, z)

Proof of Theorem isposi
StepHypRef Expression
1 isposi.k . 2  |-  K  e. 
_V
2 isposi.1 . . . . 5  |-  ( x  e.  B  ->  x  .<_  x )
323ad2ant1 979 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  x  .<_  x )
4 isposi.2 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y ) )
543adant3 978 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y ) )
6 isposi.3 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )
73, 5, 63jca 1135 . . 3  |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y )  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) )
87rgen3 2809 . 2  |-  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y )  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )
9 isposi.b . . 3  |-  B  =  ( Base `  K
)
10 isposi.l . . 3  |-  .<_  =  ( le `  K )
119, 10ispos 14435 . 2  |-  ( K  e.  Poset 
<->  ( K  e.  _V  /\ 
A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y )  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
121, 8, 11mpbir2an 888 1  |-  K  e. 
Poset
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   _Vcvv 2962   class class class wbr 4237   ` cfv 5483   Basecbs 13500   lecple 13567   Posetcpo 14428
This theorem is referenced by:  isposix  14445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-nul 4363
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-poset 14434
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