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Theorem cnvpsb 14645
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 14644 . 2  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
2 cnvps 14644 . . 3  |-  ( `' R  e.  PosetRel  ->  `' `' R  e.  PosetRel )
3 dfrel2 5321 . . . 4  |-  ( Rel 
R  <->  `' `' R  =  R
)
4 eleq1 2496 . . . . 5  |-  ( `' `' R  =  R  ->  ( `' `' R  e. 
PosetRel  <-> 
R  e.  PosetRel ) )
54biimpd 199 . . . 4  |-  ( `' `' R  =  R  ->  ( `' `' R  e. 
PosetRel  ->  R  e.  PosetRel ) )
63, 5sylbi 188 . . 3  |-  ( Rel 
R  ->  ( `' `' R  e.  PosetRel  ->  R  e. 
PosetRel ) )
72, 6syl5 30 . 2  |-  ( Rel 
R  ->  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) )
81, 7impbid2 196 1  |-  ( Rel 
R  ->  ( R  e. 
PosetRel  <->  `' R  e.  PosetRel ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   `'ccnv 4877   Rel wrel 4883   PosetRelcps 14624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ps 14629
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