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Theorem odupos 14491
Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Hypothesis
Ref Expression
odupos.d  |-  D  =  (ODual `  O )
Assertion
Ref Expression
odupos  |-  ( O  e.  Poset  ->  D  e.  Poset
)

Proof of Theorem odupos
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odupos.d . . . 4  |-  D  =  (ODual `  O )
2 fvex 5684 . . . 4  |-  (ODual `  O )  e.  _V
31, 2eqeltri 2459 . . 3  |-  D  e. 
_V
43a1i 11 . 2  |-  ( O  e.  Poset  ->  D  e.  _V )
5 eqid 2389 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
61, 5odubas 14489 . . 3  |-  ( Base `  O )  =  (
Base `  D )
76a1i 11 . 2  |-  ( O  e.  Poset  ->  ( Base `  O )  =  (
Base `  D )
)
8 eqid 2389 . . . 4  |-  ( le
`  O )  =  ( le `  O
)
91, 8oduleval 14487 . . 3  |-  `' ( le `  O )  =  ( le `  D )
109a1i 11 . 2  |-  ( O  e.  Poset  ->  `' ( le `  O )  =  ( le `  D
) )
115, 8posref 14337 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a
( le `  O
) a )
12 vex 2904 . . . 4  |-  a  e. 
_V
1312, 12brcnv 4997 . . 3  |-  ( a `' ( le `  O ) a  <->  a ( le `  O ) a )
1411, 13sylibr 204 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a `' ( le `  O ) a )
15 vex 2904 . . . . 5  |-  b  e. 
_V
1612, 15brcnv 4997 . . . 4  |-  ( a `' ( le `  O ) b  <->  b ( le `  O ) a )
1715, 12brcnv 4997 . . . 4  |-  ( b `' ( le `  O ) a  <->  a ( le `  O ) b )
1816, 17anbi12ci 680 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  <->  ( a
( le `  O
) b  /\  b
( le `  O
) a ) )
195, 8posasymb 14338 . . . 4  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  <-> 
a  =  b ) )
2019biimpd 199 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  ->  a  =  b ) )
2118, 20syl5bi 209 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  ->  a  =  b ) )
22 3anrev 947 . . . 4  |-  ( ( a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
)  <->  ( c  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )  /\  a  e.  ( Base `  O ) ) )
235, 8postr 14339 . . . 4  |-  ( ( O  e.  Poset  /\  (
c  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  a  e.  ( Base `  O )
) )  ->  (
( c ( le
`  O ) b  /\  b ( le
`  O ) a )  ->  c ( le `  O ) a ) )
2422, 23sylan2b 462 . . 3  |-  ( ( O  e.  Poset  /\  (
a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
) )  ->  (
( c ( le
`  O ) b  /\  b ( le
`  O ) a )  ->  c ( le `  O ) a ) )
25 vex 2904 . . . . 5  |-  c  e. 
_V
2615, 25brcnv 4997 . . . 4  |-  ( b `' ( le `  O ) c  <->  c ( le `  O ) b )
2716, 26anbi12ci 680 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) c )  <->  ( c
( le `  O
) b  /\  b
( le `  O
) a ) )
2812, 25brcnv 4997 . . 3  |-  ( a `' ( le `  O ) c  <->  c ( le `  O ) a )
2924, 27, 283imtr4g 262 . 2  |-  ( ( O  e.  Poset  /\  (
a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
) )  ->  (
( a `' ( le `  O ) b  /\  b `' ( le `  O
) c )  -> 
a `' ( le
`  O ) c ) )
304, 7, 10, 14, 21, 29isposd 14341 1  |-  ( O  e.  Poset  ->  D  e.  Poset
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901   class class class wbr 4155   `'ccnv 4819   ` cfv 5396   Basecbs 13398   lecple 13465   Posetcpo 14326  ODualcodu 14484
This theorem is referenced by:  oduposb  14492  posglbd  14505
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ple 13478  df-poset 14332  df-odu 14485
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