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Theorem pocnv 30398
Description: The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
pocnv  |-  ( R  Po  A  ->  `' R  Po  A )

Proof of Theorem pocnv
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poirr 4781 . . 3  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
2 vex 3084 . . . 4  |-  x  e. 
_V
32, 2brcnv 5032 . . 3  |-  ( x `' R x  <->  x R x )
41, 3sylnibr 306 . 2  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x `' R x )
5 3anrev 993 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( z  e.  A  /\  y  e.  A  /\  x  e.  A )
)
6 potr 4782 . . . 4  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  y  e.  A  /\  x  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
75, 6sylan2b 477 . . 3  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
8 vex 3084 . . . . 5  |-  y  e. 
_V
92, 8brcnv 5032 . . . 4  |-  ( x `' R y  <->  y R x )
10 vex 3084 . . . . 5  |-  z  e. 
_V
118, 10brcnv 5032 . . . 4  |-  ( y `' R z  <->  z R
y )
129, 11anbi12ci 702 . . 3  |-  ( ( x `' R y  /\  y `' R
z )  <->  ( z R y  /\  y R x ) )
132, 10brcnv 5032 . . 3  |-  ( x `' R z  <->  z R x )
147, 12, 133imtr4g 273 . 2  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x `' R
y  /\  y `' R z )  ->  x `' R z ) )
154, 14ispod 4778 1  |-  ( R  Po  A  ->  `' R  Po  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1868   class class class wbr 4420    Po wpo 4768   `'ccnv 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-po 4770  df-cnv 4857
This theorem is referenced by:  socnv  30399
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