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Theorem pocnv 29437
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 4800 . . 3  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
2 vex 3109 . . . 4  |-  x  e. 
_V
32, 2brcnv 5174 . . 3  |-  ( x `' R x  <->  x R x )
41, 3sylnibr 303 . 2  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x `' R x )
5 3anrev 982 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( z  e.  A  /\  y  e.  A  /\  x  e.  A )
)
6 potr 4801 . . . 4  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  y  e.  A  /\  x  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
75, 6sylan2b 473 . . 3  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( z R y  /\  y R x )  ->  z R x ) )
8 vex 3109 . . . . 5  |-  y  e. 
_V
92, 8brcnv 5174 . . . 4  |-  ( x `' R y  <->  y R x )
10 vex 3109 . . . . 5  |-  z  e. 
_V
118, 10brcnv 5174 . . . 4  |-  ( y `' R z  <->  z R
y )
129, 11anbi12ci 696 . . 3  |-  ( ( x `' R y  /\  y `' R
z )  <->  ( z R y  /\  y R x ) )
132, 10brcnv 5174 . . 3  |-  ( x `' R z  <->  z R x )
147, 12, 133imtr4g 270 . 2  |-  ( ( R  Po  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x `' R
y  /\  y `' R z )  ->  x `' R z ) )
154, 14ispod 4797 1  |-  ( R  Po  A  ->  `' R  Po  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1823   class class class wbr 4439    Po wpo 4787   `'ccnv 4987
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-po 4789  df-cnv 4996
This theorem is referenced by:  socnv  29438
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