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Theorem cnvso 6308
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 6307 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
2 ralcom 3289 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
3 vex 3484 . . . . . . 7 𝑦 ∈ V
4 vex 3484 . . . . . . 7 𝑥 ∈ V
53, 4brcnv 5893 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
6 equcom 2017 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
74, 3brcnv 5893 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 6, 73orbi123i 1157 . . . . 5 ((𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
982ralbii 3128 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
102, 9bitr4i 278 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦))
111, 10anbi12i 628 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
12 df-so 5593 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
13 df-so 5593 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
1411, 12, 133bitr4i 303 1 (𝑅 Or 𝐴𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3o 1086  wral 3061   class class class wbr 5143   Po wpo 5590   Or wor 5591  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693
This theorem is referenced by:  infexd  9523  eqinf  9524  infval  9526  infcl  9528  inflb  9529  infglb  9530  infglbb  9531  fiinfcl  9541  infltoreq  9542  infempty  9547  infiso  9548  wofib  9585  oemapso  9722  cflim2  10303  fin23lem40  10391  gtso  11342  nomaxmo  27743  tosglb  32965  xrsclat  33013  xrge0iifiso  33934  socnv  35764  welb  37743  xrgtso  45356
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