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Theorem cnvsom 4804
Description: The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
Assertion
Ref Expression
cnvsom (x x A → (𝑅 Or A𝑅 Or A))
Distinct variable groups:   x,A   x,𝑅

Proof of Theorem cnvsom
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvpom 4803 . . 3 (x x A → (𝑅 Po A𝑅 Po A))
2 vex 2554 . . . . . . . . 9 y V
3 vex 2554 . . . . . . . . 9 x V
42, 3brcnv 4461 . . . . . . . 8 (y𝑅xx𝑅y)
5 vex 2554 . . . . . . . . . . 11 z V
62, 5brcnv 4461 . . . . . . . . . 10 (y𝑅zz𝑅y)
75, 3brcnv 4461 . . . . . . . . . 10 (z𝑅xx𝑅z)
86, 7orbi12i 680 . . . . . . . . 9 ((y𝑅z z𝑅x) ↔ (z𝑅y x𝑅z))
9 orcom 646 . . . . . . . . 9 ((z𝑅y x𝑅z) ↔ (x𝑅z z𝑅y))
108, 9bitri 173 . . . . . . . 8 ((y𝑅z z𝑅x) ↔ (x𝑅z z𝑅y))
114, 10imbi12i 228 . . . . . . 7 ((y𝑅x → (y𝑅z z𝑅x)) ↔ (x𝑅y → (x𝑅z z𝑅y)))
1211ralbii 2324 . . . . . 6 (z A (y𝑅x → (y𝑅z z𝑅x)) ↔ z A (x𝑅y → (x𝑅z z𝑅y)))
13122ralbii 2326 . . . . 5 (x A y A z A (y𝑅x → (y𝑅z z𝑅x)) ↔ x A y A z A (x𝑅y → (x𝑅z z𝑅y)))
14 ralcom 2467 . . . . 5 (x A y A z A (y𝑅x → (y𝑅z z𝑅x)) ↔ y A x A z A (y𝑅x → (y𝑅z z𝑅x)))
1513, 14bitr3i 175 . . . 4 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ y A x A z A (y𝑅x → (y𝑅z z𝑅x)))
1615a1i 9 . . 3 (x x A → (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ y A x A z A (y𝑅x → (y𝑅z z𝑅x))))
171, 16anbi12d 442 . 2 (x x A → ((𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))) ↔ (𝑅 Po A y A x A z A (y𝑅x → (y𝑅z z𝑅x)))))
18 df-iso 4025 . 2 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
19 df-iso 4025 . 2 (𝑅 Or A ↔ (𝑅 Po A y A x A z A (y𝑅x → (y𝑅z z𝑅x))))
2017, 18, 193bitr4g 212 1 (x x A → (𝑅 Or A𝑅 Or A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  wex 1378   wcel 1390  wral 2300   class class class wbr 3755   Po wpo 4022   Or wor 4023  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-po 4024  df-iso 4025  df-cnv 4296
This theorem is referenced by:  gtso  6854
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