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Theorem isposi 14372
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 978 . . . 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 977 . . . 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 1134 . . 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 2767 . 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 14363 . 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 887 1  |-  K  e. 
Poset
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   class class class wbr 4176   ` cfv 5417   Basecbs 13428   lecple 13495   Posetcpo 14356
This theorem is referenced by:  isposix  14373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-nul 4302
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-poset 14362
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