Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pocnv Structured version   Visualization version   GIF version

Theorem pocnv 31354
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 𝐴𝑅 Po 𝐴)

Proof of Theorem pocnv
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poirr 5011 . . 3 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
2 vex 3194 . . . 4 𝑥 ∈ V
32, 2brcnv 5270 . . 3 (𝑥𝑅𝑥𝑥𝑅𝑥)
41, 3sylnibr 319 . 2 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 3anrev 1047 . . . 4 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑧𝐴𝑦𝐴𝑥𝐴))
6 potr 5012 . . . 4 ((𝑅 Po 𝐴 ∧ (𝑧𝐴𝑦𝐴𝑥𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
75, 6sylan2b 492 . . 3 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8 vex 3194 . . . . 5 𝑦 ∈ V
92, 8brcnv 5270 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3194 . . . . 5 𝑧 ∈ V
118, 10brcnv 5270 . . . 4 (𝑦𝑅𝑧𝑧𝑅𝑦)
129, 11anbi12ci 733 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑧𝑅𝑦𝑦𝑅𝑥))
132, 10brcnv 5270 . . 3 (𝑥𝑅𝑧𝑧𝑅𝑥)
147, 12, 133imtr4g 285 . 2 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 5008 1 (𝑅 Po 𝐴𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1992   class class class wbr 4618   Po wpo 4998  ccnv 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-po 5000  df-cnv 5087
This theorem is referenced by:  socnv  31355
  Copyright terms: Public domain W3C validator