MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvso Structured version   Visualization version   GIF version

Theorem cnvso 6284
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 6283 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
2 ralcom 3286 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
3 vex 3478 . . . . . . 7 𝑦 ∈ V
4 vex 3478 . . . . . . 7 𝑥 ∈ V
53, 4brcnv 5880 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
6 equcom 2021 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
74, 3brcnv 5880 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 6, 73orbi123i 1156 . . . . 5 ((𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
982ralbii 3128 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
102, 9bitr4i 277 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦))
111, 10anbi12i 627 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
12 df-so 5588 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
13 df-so 5588 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
1411, 12, 133bitr4i 302 1 (𝑅 Or 𝐴𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3o 1086  wral 3061   class class class wbr 5147   Po wpo 5585   Or wor 5586  ccnv 5674
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-po 5587  df-so 5588  df-cnv 5683
This theorem is referenced by:  infexd  9474  eqinf  9475  infval  9477  infcl  9479  inflb  9480  infglb  9481  infglbb  9482  fiinfcl  9492  infltoreq  9493  infempty  9498  infiso  9499  wofib  9536  oemapso  9673  cflim2  10254  fin23lem40  10342  gtso  11291  nomaxmo  27190  tosglb  32132  xrsclat  32168  xrge0iifiso  32903  socnv  34722  welb  36592  xrgtso  44041
  Copyright terms: Public domain W3C validator