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Theorem pocnv 33006
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 5458 . . 3 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
2 vex 3474 . . . 4 𝑥 ∈ V
32, 2brcnv 5726 . . 3 (𝑥𝑅𝑥𝑥𝑅𝑥)
41, 3sylnibr 332 . 2 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 3anrev 1098 . . . 4 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑧𝐴𝑦𝐴𝑥𝐴))
6 potr 5459 . . . 4 ((𝑅 Po 𝐴 ∧ (𝑧𝐴𝑦𝐴𝑥𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
75, 6sylan2b 596 . . 3 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8 vex 3474 . . . . 5 𝑦 ∈ V
92, 8brcnv 5726 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3474 . . . . 5 𝑧 ∈ V
118, 10brcnv 5726 . . . 4 (𝑦𝑅𝑧𝑧𝑅𝑦)
129, 11anbi12ci 630 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑧𝑅𝑦𝑦𝑅𝑥))
132, 10brcnv 5726 . . 3 (𝑥𝑅𝑧𝑧𝑅𝑥)
147, 12, 133imtr4g 299 . 2 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 5455 1 (𝑅 Po 𝐴𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115   class class class wbr 5039   Po wpo 5445  ccnv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-po 5447  df-cnv 5536
This theorem is referenced by: (None)
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