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Theorem cnvso 6111
Description: The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
Assertion
Ref Expression
cnvso (𝑅 Or 𝐴𝑅 Or 𝐴)

Proof of Theorem cnvso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvpo 6110 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
2 ralcom 3310 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
3 vex 3447 . . . . . . 7 𝑦 ∈ V
4 vex 3447 . . . . . . 7 𝑥 ∈ V
53, 4brcnv 5721 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
6 equcom 2025 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
74, 3brcnv 5721 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 6, 73orbi123i 1153 . . . . 5 ((𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
982ralbii 3137 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
102, 9bitr4i 281 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦))
111, 10anbi12i 629 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
12 df-so 5443 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
13 df-so 5443 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
1411, 12, 133bitr4i 306 1 (𝑅 Or 𝐴𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3o 1083  wral 3109   class class class wbr 5033   Po wpo 5440   Or wor 5441  ccnv 5522
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-po 5442  df-so 5443  df-cnv 5531
This theorem is referenced by:  infexd  8935  eqinf  8936  infval  8938  infcl  8940  inflb  8941  infglb  8942  infglbb  8943  fiinfcl  8953  infltoreq  8954  infempty  8959  infiso  8960  wofib  8997  oemapso  9133  cflim2  9678  fin23lem40  9766  gtso  10715  tosglb  30686  xrsclat  30717  xrge0iifiso  31286  socnv  33108  nomaxmo  33309  welb  35167  xrgtso  41964
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