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	|- (E.x x e. A -> ( R Or A <-> `' R 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.

Step	Hyp	Ref	Expression
1		cnvpom 4803	. . 3 |- (E.x x e. A -> ( R Po A <-> `' R Po A))
2		vex 2554	. . . . . . . . 9 |- y e. _V
3		vex 2554	. . . . . . . . 9 |- x e. _V
4	2, 3	brcnv 4461	. . . . . . . 8 |- (y `' Rx <-> x Ry)
5		vex 2554	. . . . . . . . . . 11 |- z e. _V
6	2, 5	brcnv 4461	. . . . . . . . . 10 |- (y `' Rz <-> z Ry)
7	5, 3	brcnv 4461	. . . . . . . . . 10 |- (z `' Rx <-> x Rz)
8	6, 7	orbi12i 680	. . . . . . . . 9 |- ((y `' Rz \/ z `' Rx) <-> (z Ry \/ x Rz))
9		orcom 646	. . . . . . . . 9 |- ((z Ry \/ x Rz) <-> (x Rz \/ z Ry))
10	8, 9	bitri 173	. . . . . . . 8 |- ((y `' Rz \/ z `' Rx) <-> (x Rz \/ z Ry))
11	4, 10	imbi12i 228	. . . . . . 7 |- ((y `' Rx -> (y `' Rz \/ z `' Rx)) <-> (x Ry -> (x Rz \/ z Ry)))
12	11	ralbii 2324	. . . . . 6 |- (A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)) <-> A.z e. A (x Ry -> (x Rz \/ z Ry)))
13	12	2ralbii 2326	. . . . 5 |- (A.x e. A A.y e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)) <-> A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)))
14		ralcom 2467	. . . . 5 |- (A.x e. A A.y e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)) <-> A.y e. A A.x e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)))
15	13, 14	bitr3i 175	. . . 4 |- (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) <-> A.y e. A A.x e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)))
16	15	a1i 9	. . 3 |- (E.x x e. A -> (A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry)) <-> A.y e. A A.x e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx))))
17	1, 16	anbi12d 442	. 2 |- (E.x x e. A -> (( R Po A /\ A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry))) <-> ( `' R Po A /\ A.y e. A A.x e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx)))))
18		df-iso 4025	. 2 |- ( R Or A <-> ( R Po A /\ A.x e. A A.y e. A A.z e. A (x Ry -> (x Rz \/ z Ry))))
19		df-iso 4025	. 2 |- ( `' R Or A <-> ( `' R Po A /\ A.y e. A A.x e. A A.z e. A (y `' Rx -> (y `' Rz \/ z `' Rx))))
20	17, 18, 19	3bitr4g 212	1 |- (E.x x e. A -> ( R Or A <-> `' R Or A))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  E.wex 1378   e. wcel 1390  A.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-bndl 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  6894