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Theorem pocnv 30398
 Description: The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
pocnv

Proof of Theorem pocnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poirr 4781 . . 3
2 vex 3084 . . . 4
32, 2brcnv 5032 . . 3
41, 3sylnibr 306 . 2
5 3anrev 993 . . . 4
6 potr 4782 . . . 4
75, 6sylan2b 477 . . 3
8 vex 3084 . . . . 5
92, 8brcnv 5032 . . . 4
10 vex 3084 . . . . 5
118, 10brcnv 5032 . . . 4
129, 11anbi12ci 702 . . 3
132, 10brcnv 5032 . . 3
147, 12, 133imtr4g 273 . 2
154, 14ispod 4778 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wcel 1868   class class class wbr 4420   wpo 4768  ccnv 4848 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-po 4770  df-cnv 4857 This theorem is referenced by:  socnv  30399
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