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Theorem socnv 30908
 Description: The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
socnv (𝑅 Or 𝐴𝑅 Or 𝐴)

Proof of Theorem socnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4976 . . 3 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 pocnv 30907 . . 3 (𝑅 Po 𝐴𝑅 Po 𝐴)
31, 2syl 17 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
4 solin 4982 . . 3 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
5 vex 3176 . . . . . 6 𝑥 ∈ V
6 vex 3176 . . . . . 6 𝑦 ∈ V
75, 6brcnv 5227 . . . . 5 (𝑥𝑅𝑦𝑦𝑅𝑥)
8 biid 250 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
96, 5brcnv 5227 . . . . 5 (𝑦𝑅𝑥𝑥𝑅𝑦)
107, 8, 93orbi123i 1245 . . . 4 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥 = 𝑦𝑥𝑅𝑦))
11 orcom 401 . . . . . 6 ((𝑦𝑅𝑥𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
1211orbi2i 540 . . . . 5 ((𝑥 = 𝑦 ∨ (𝑦𝑅𝑥𝑥𝑅𝑦)) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦𝑦𝑅𝑥)))
13 3orass 1034 . . . . . 6 ((𝑦𝑅𝑥𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑦𝑅𝑥 ∨ (𝑥 = 𝑦𝑥𝑅𝑦)))
14 or12 544 . . . . . 6 ((𝑦𝑅𝑥 ∨ (𝑥 = 𝑦𝑥𝑅𝑦)) ↔ (𝑥 = 𝑦 ∨ (𝑦𝑅𝑥𝑥𝑅𝑦)))
1513, 14bitri 263 . . . . 5 ((𝑦𝑅𝑥𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ (𝑦𝑅𝑥𝑥𝑅𝑦)))
16 3orass 1034 . . . . . 6 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
17 or12 544 . . . . . 6 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦𝑦𝑅𝑥)))
1816, 17bitri 263 . . . . 5 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦𝑦𝑅𝑥)))
1912, 15, 183bitr4i 291 . . . 4 ((𝑦𝑅𝑥𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2010, 19bitri 263 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
214, 20sylibr 223 . 2 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
223, 21issod 4989 1 (𝑅 Or 𝐴𝑅 Or 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∈ wcel 1977   class class class wbr 4583   Po wpo 4957   Or wor 4958  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-cnv 5046 This theorem is referenced by:  wzelOLD  31016
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