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Theorem cnvso 5633
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 5632 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
2 ralcom 3090 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
3 vex 3189 . . . . . . 7 𝑦 ∈ V
4 vex 3189 . . . . . . 7 𝑥 ∈ V
53, 4brcnv 5265 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
6 equcom 1942 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
74, 3brcnv 5265 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 6, 73orbi123i 1250 . . . . 5 ((𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
982ralbii 2975 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
102, 9bitr4i 267 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦))
111, 10anbi12i 732 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
12 df-so 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
13 df-so 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥𝑦 = 𝑥𝑥𝑅𝑦)))
1411, 12, 133bitr4i 292 1 (𝑅 Or 𝐴𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3o 1035  wral 2907   class class class wbr 4613   Po wpo 4993   Or wor 4994  ccnv 5073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-po 4995  df-so 4996  df-cnv 5082
This theorem is referenced by:  infexd  8333  eqinf  8334  infval  8336  infcl  8338  inflb  8339  infglb  8340  infglbb  8341  fiinfcl  8351  infltoreq  8352  infempty  8356  infiso  8357  wofib  8394  oemapso  8523  cflim2  9029  fin23lem40  9117  gtso  10063  tosglb  29452  xrsclat  29462  xrge0iifiso  29760  inffzOLD  31320  welb  33160  xrgtso  39022
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