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Theorem pocnv 27708
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 4750 . . 3  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
2 vex 3071 . . . . 5  |-  x  e. 
_V
32, 2brcnv 5120 . . . 4  |-  ( x `' R x  <->  x R x )
43notbii 296 . . 3  |-  ( -.  x `' R x  <->  -.  x R x )
51, 4sylibr 212 . 2  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x `' R x )
6 3anrev 976 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( z  e.  A  /\  y  e.  A  /\  x  e.  A )
)
7 potr 4751 . . . 4  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  y  e.  A  /\  x  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
86, 7sylan2b 475 . . 3  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
9 ancom 450 . . . 4  |-  ( ( x `' R y  /\  y `' R
z )  <->  ( y `' R z  /\  x `' R y ) )
10 vex 3071 . . . . . 6  |-  y  e. 
_V
11 vex 3071 . . . . . 6  |-  z  e. 
_V
1210, 11brcnv 5120 . . . . 5  |-  ( y `' R z  <->  z R
y )
132, 10brcnv 5120 . . . . 5  |-  ( x `' R y  <->  y R x )
1412, 13anbi12i 697 . . . 4  |-  ( ( y `' R z  /\  x `' R
y )  <->  ( z R y  /\  y R x ) )
159, 14bitri 249 . . 3  |-  ( ( x `' R y  /\  y `' R
z )  <->  ( z R y  /\  y R x ) )
162, 11brcnv 5120 . . 3  |-  ( x `' R z  <->  z R x )
178, 15, 163imtr4g 270 . 2  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x `' R
y  /\  y `' R z )  ->  x `' R z ) )
185, 17ispod 4747 1  |-  ( R  Po  A  ->  `' R  Po  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   class class class wbr 4390    Po wpo 4737   `'ccnv 4937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-po 4739  df-cnv 4946
This theorem is referenced by:  socnv  27709
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