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Theorem pocnv 32167
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 5244 . . 3 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
2 vex 3388 . . . 4 𝑥 ∈ V
32, 2brcnv 5508 . . 3 (𝑥𝑅𝑥𝑥𝑅𝑥)
41, 3sylnibr 321 . 2 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 3anrev 1127 . . . 4 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ (𝑧𝐴𝑦𝐴𝑥𝐴))
6 potr 5245 . . . 4 ((𝑅 Po 𝐴 ∧ (𝑧𝐴𝑦𝐴𝑥𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
75, 6sylan2b 588 . . 3 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8 vex 3388 . . . . 5 𝑦 ∈ V
92, 8brcnv 5508 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 vex 3388 . . . . 5 𝑧 ∈ V
118, 10brcnv 5508 . . . 4 (𝑦𝑅𝑧𝑧𝑅𝑦)
129, 11anbi12ci 622 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑧𝑅𝑦𝑦𝑅𝑥))
132, 10brcnv 5508 . . 3 (𝑥𝑅𝑧𝑧𝑅𝑥)
147, 12, 133imtr4g 288 . 2 ((𝑅 Po 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 5241 1 (𝑅 Po 𝐴𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wcel 2157   class class class wbr 4843   Po wpo 5231  ccnv 5311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-po 5233  df-cnv 5320
This theorem is referenced by: (None)
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