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Theorem cnvpsb 16165
Description: The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
cnvpsb  |-  ( Rel 
R  ->  ( R  e. 
PosetRel  <->  `' R  e.  PosetRel ) )

Proof of Theorem cnvpsb
StepHypRef Expression
1 cnvps 16164 . 2  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
2 cnvps 16164 . . 3  |-  ( `' R  e.  PosetRel  ->  `' `' R  e.  PosetRel )
3 dfrel2 5273 . . . 4  |-  ( Rel 
R  <->  `' `' R  =  R
)
4 eleq1 2474 . . . . 5  |-  ( `' `' R  =  R  ->  ( `' `' R  e. 
PosetRel  <-> 
R  e.  PosetRel ) )
54biimpd 207 . . . 4  |-  ( `' `' R  =  R  ->  ( `' `' R  e. 
PosetRel  ->  R  e.  PosetRel ) )
63, 5sylbi 195 . . 3  |-  ( Rel 
R  ->  ( `' `' R  e.  PosetRel  ->  R  e. 
PosetRel ) )
72, 6syl5 30 . 2  |-  ( Rel 
R  ->  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) )
81, 7impbid2 204 1  |-  ( Rel 
R  ->  ( R  e. 
PosetRel  <->  `' R  e.  PosetRel ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   `'ccnv 4821   Rel wrel 4827   PosetRelcps 16150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ps 16152
This theorem is referenced by: (None)
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