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Theorem odupos 14239
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 5539 . . . 4  |-  (ODual `  O )  e.  _V
31, 2eqeltri 2353 . . 3  |-  D  e. 
_V
43a1i 10 . 2  |-  ( O  e.  Poset  ->  D  e.  _V )
5 eqid 2283 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
61, 5odubas 14237 . . 3  |-  ( Base `  O )  =  (
Base `  D )
76a1i 10 . 2  |-  ( O  e.  Poset  ->  ( Base `  O )  =  (
Base `  D )
)
8 eqid 2283 . . . 4  |-  ( le
`  O )  =  ( le `  O
)
91, 8oduleval 14235 . . 3  |-  `' ( le `  O )  =  ( le `  D )
109a1i 10 . 2  |-  ( O  e.  Poset  ->  `' ( le `  O )  =  ( le `  D
) )
115, 8posref 14085 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a
( le `  O
) a )
12 vex 2791 . . . 4  |-  a  e. 
_V
1312, 12brcnv 4864 . . 3  |-  ( a `' ( le `  O ) a  <->  a ( le `  O ) a )
1411, 13sylibr 203 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a `' ( le `  O ) a )
15 vex 2791 . . . . 5  |-  b  e. 
_V
1612, 15brcnv 4864 . . . 4  |-  ( a `' ( le `  O ) b  <->  b ( le `  O ) a )
1715, 12brcnv 4864 . . . 4  |-  ( b `' ( le `  O ) a  <->  a ( le `  O ) b )
1816, 17anbi12ci 679 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  <->  ( a
( le `  O
) b  /\  b
( le `  O
) a ) )
195, 8posasymb 14086 . . . 4  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  <-> 
a  =  b ) )
2019biimpd 198 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  ->  a  =  b ) )
2118, 20syl5bi 208 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  ->  a  =  b ) )
22 3anrev 945 . . . 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 14087 . . . 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 461 . . 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 2791 . . . . 5  |-  c  e. 
_V
2615, 25brcnv 4864 . . . 4  |-  ( b `' ( le `  O ) c  <->  c ( le `  O ) b )
2716, 26anbi12ci 679 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) c )  <->  ( c
( le `  O
) b  /\  b
( le `  O
) a ) )
2812, 25brcnv 4864 . . 3  |-  ( a `' ( le `  O ) c  <->  c ( le `  O ) a )
2924, 27, 283imtr4g 261 . 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 14089 1  |-  ( O  e.  Poset  ->  D  e.  Poset
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   `'ccnv 4688   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074  ODualcodu 14232
This theorem is referenced by:  oduposb  14240  posglbd  14253
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ple 13228  df-poset 14080  df-odu 14233
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