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Theorem cnvso 6239
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 6238 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
2 ralcom 3271 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
3 vex 3448 . . . . . . 7 𝑦 ∈ V
4 vex 3448 . . . . . . 7 𝑥 ∈ V
53, 4brcnv 5837 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
6 equcom 2022 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
74, 3brcnv 5837 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 6, 73orbi123i 1157 . . . . 5 ((𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
982ralbii 3126 . . . 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 5545 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
13 df-so 5545 . 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 205  wa 397  w3o 1087  wral 3063   class class class wbr 5104   Po wpo 5542   Or wor 5543  ccnv 5631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-po 5544  df-so 5545  df-cnv 5640
This theorem is referenced by:  infexd  9378  eqinf  9379  infval  9381  infcl  9383  inflb  9384  infglb  9385  infglbb  9386  fiinfcl  9396  infltoreq  9397  infempty  9402  infiso  9403  wofib  9440  oemapso  9577  cflim2  10158  fin23lem40  10246  gtso  11195  nomaxmo  26998  tosglb  31660  xrsclat  31696  xrge0iifiso  32320  socnv  34147  welb  36127  xrgtso  43478
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