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Theorem odupos 14255
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 5555 . . . 4  |-  (ODual `  O )  e.  _V
31, 2eqeltri 2366 . . 3  |-  D  e. 
_V
43a1i 10 . 2  |-  ( O  e.  Poset  ->  D  e.  _V )
5 eqid 2296 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
61, 5odubas 14253 . . 3  |-  ( Base `  O )  =  (
Base `  D )
76a1i 10 . 2  |-  ( O  e.  Poset  ->  ( Base `  O )  =  (
Base `  D )
)
8 eqid 2296 . . . 4  |-  ( le
`  O )  =  ( le `  O
)
91, 8oduleval 14251 . . 3  |-  `' ( le `  O )  =  ( le `  D )
109a1i 10 . 2  |-  ( O  e.  Poset  ->  `' ( le `  O )  =  ( le `  D
) )
115, 8posref 14101 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a
( le `  O
) a )
12 vex 2804 . . . 4  |-  a  e. 
_V
1312, 12brcnv 4880 . . 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 2804 . . . . 5  |-  b  e. 
_V
1612, 15brcnv 4880 . . . 4  |-  ( a `' ( le `  O ) b  <->  b ( le `  O ) a )
1715, 12brcnv 4880 . . . 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 14102 . . . 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 14103 . . . 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 2804 . . . . 5  |-  c  e. 
_V
2615, 25brcnv 4880 . . . 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 4880 . . 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 14105 1  |-  ( O  e.  Poset  ->  D  e.  Poset
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   `'ccnv 4704   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090  ODualcodu 14248
This theorem is referenced by:  oduposb  14256  posglbd  14269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ple 13244  df-poset 14096  df-odu 14249
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