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Theorem odupos 15964
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 5858 . . . 4  |-  (ODual `  O )  e.  _V
31, 2eqeltri 2538 . . 3  |-  D  e. 
_V
43a1i 11 . 2  |-  ( O  e.  Poset  ->  D  e.  _V )
5 eqid 2454 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
61, 5odubas 15962 . . 3  |-  ( Base `  O )  =  (
Base `  D )
76a1i 11 . 2  |-  ( O  e.  Poset  ->  ( Base `  O )  =  (
Base `  D )
)
8 eqid 2454 . . . 4  |-  ( le
`  O )  =  ( le `  O
)
91, 8oduleval 15960 . . 3  |-  `' ( le `  O )  =  ( le `  D )
109a1i 11 . 2  |-  ( O  e.  Poset  ->  `' ( le `  O )  =  ( le `  D
) )
115, 8posref 15779 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a
( le `  O
) a )
12 vex 3109 . . . 4  |-  a  e. 
_V
1312, 12brcnv 5174 . . 3  |-  ( a `' ( le `  O ) a  <->  a ( le `  O ) a )
1411, 13sylibr 212 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a `' ( le `  O ) a )
15 vex 3109 . . . . 5  |-  b  e. 
_V
1612, 15brcnv 5174 . . . 4  |-  ( a `' ( le `  O ) b  <->  b ( le `  O ) a )
1715, 12brcnv 5174 . . . 4  |-  ( b `' ( le `  O ) a  <->  a ( le `  O ) b )
1816, 17anbi12ci 696 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  <->  ( a
( le `  O
) b  /\  b
( le `  O
) a ) )
195, 8posasymb 15781 . . . 4  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  <-> 
a  =  b ) )
2019biimpd 207 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  ->  a  =  b ) )
2118, 20syl5bi 217 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  ->  a  =  b ) )
22 3anrev 982 . . . 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 15782 . . . 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 473 . . 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 3109 . . . . 5  |-  c  e. 
_V
2615, 25brcnv 5174 . . . 4  |-  ( b `' ( le `  O ) c  <->  c ( le `  O ) b )
2716, 26anbi12ci 696 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) c )  <->  ( c
( le `  O
) b  /\  b
( le `  O
) a ) )
2812, 25brcnv 5174 . . 3  |-  ( a `' ( le `  O ) c  <->  c ( le `  O ) a )
2924, 27, 283imtr4g 270 . 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 15784 1  |-  ( O  e.  Poset  ->  D  e.  Poset
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439   `'ccnv 4987   ` cfv 5570   Basecbs 14716   lecple 14791   Posetcpo 15768  ODualcodu 15957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ple 14804  df-preset 15756  df-poset 15774  df-odu 15958
This theorem is referenced by:  oduposb  15965  posglbd  15979  odutos  27885
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