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Theorem pocnv 30907
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 4970 . . 3 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
2 vex 3176 . . . 4 𝑥 ∈ V
32, 2brcnv 5227 . . 3 (𝑥𝑅𝑥𝑥𝑅𝑥)
41, 3sylnibr 318 . 2 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 3anrev 1042 . . . 4 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑧𝐴𝑦𝐴𝑥𝐴))
6 potr 4971 . . . 4 ((𝑅 Po 𝐴 ∧ (𝑧𝐴𝑦𝐴𝑥𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
75, 6sylan2b 491 . . 3 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8 vex 3176 . . . . 5 𝑦 ∈ V
92, 8brcnv 5227 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3176 . . . . 5 𝑧 ∈ V
118, 10brcnv 5227 . . . 4 (𝑦𝑅𝑧𝑧𝑅𝑦)
129, 11anbi12ci 730 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑧𝑅𝑦𝑦𝑅𝑥))
132, 10brcnv 5227 . . 3 (𝑥𝑅𝑧𝑧𝑅𝑥)
147, 12, 133imtr4g 284 . 2 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 4967 1 (𝑅 Po 𝐴𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wcel 1977   class class class wbr 4583   Po wpo 4957  ccnv 5037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-cnv 5046
This theorem is referenced by:  socnv  30908
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