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Theorem pocnv 33449
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 5480 . . 3 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
2 vex 3412 . . . 4 𝑥 ∈ V
32, 2brcnv 5751 . . 3 (𝑥𝑅𝑥𝑥𝑅𝑥)
41, 3sylnibr 332 . 2 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 3anrev 1103 . . . 4 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑧𝐴𝑦𝐴𝑥𝐴))
6 potr 5481 . . . 4 ((𝑅 Po 𝐴 ∧ (𝑧𝐴𝑦𝐴𝑥𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
75, 6sylan2b 597 . . 3 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8 vex 3412 . . . . 5 𝑦 ∈ V
92, 8brcnv 5751 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3412 . . . . 5 𝑧 ∈ V
118, 10brcnv 5751 . . . 4 (𝑦𝑅𝑧𝑧𝑅𝑦)
129, 11anbi12ci 631 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑧𝑅𝑦𝑦𝑅𝑥))
132, 10brcnv 5751 . . 3 (𝑥𝑅𝑧𝑧𝑅𝑥)
147, 12, 133imtr4g 299 . 2 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 5477 1 (𝑅 Po 𝐴𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089  wcel 2110   class class class wbr 5053   Po wpo 5466  ccnv 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-po 5468  df-cnv 5559
This theorem is referenced by: (None)
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