Theorem cnvsom 4803

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 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 4802	. . 3 ⊢ (∃x x ∈ A → (𝑅 Po A ↔ ◡𝑅 Po A))
2		vex 2554	. . . . . . . . 9 ⊢ y ∈ V
3		vex 2554	. . . . . . . . 9 ⊢ x ∈ V
4	2, 3	brcnv 4460	. . . . . . . 8 ⊢ (y◡𝑅x ↔ x𝑅y)
5		vex 2554	. . . . . . . . . . 11 ⊢ z ∈ V
6	2, 5	brcnv 4460	. . . . . . . . . 10 ⊢ (y◡𝑅z ↔ z𝑅y)
7	5, 3	brcnv 4460	. . . . . . . . . 10 ⊢ (z◡𝑅x ↔ x𝑅z)
8	6, 7	orbi12i 680	. . . . . . . . 9 ⊢ ((y◡𝑅z ∨ z◡𝑅x) ↔ (z𝑅y ∨ x𝑅z))
9		orcom 646	. . . . . . . . 9 ⊢ ((z𝑅y ∨ x𝑅z) ↔ (x𝑅z ∨ z𝑅y))
10	8, 9	bitri 173	. . . . . . . 8 ⊢ ((y◡𝑅z ∨ z◡𝑅x) ↔ (x𝑅z ∨ z𝑅y))
11	4, 10	imbi12i 228	. . . . . . 7 ⊢ ((y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)) ↔ (x𝑅y → (x𝑅z ∨ z𝑅y)))
12	11	ralbii 2324	. . . . . 6 ⊢ (∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)) ↔ ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)))
13	12	2ralbii 2326	. . . . 5 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)) ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)))
14		ralcom 2467	. . . . 5 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)) ↔ ∀y ∈ A ∀x ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)))
15	13, 14	bitr3i 175	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀y ∈ A ∀x ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)))
16	15	a1i 9	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀y ∈ A ∀x ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x))))
17	1, 16	anbi12d 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 (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x)))))
18		df-iso 4024	. 2 ⊢ (𝑅 Or A ↔ (𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))))
19		df-iso 4024	. 2 ⊢ (◡𝑅 Or A ↔ (◡𝑅 Po A ∧ ∀y ∈ A ∀x ∈ A ∀z ∈ A (y◡𝑅x → (y◡𝑅z ∨ z◡𝑅x))))
20	17, 18, 19	3bitr4g 212	1 ⊢ (∃x x ∈ A → (𝑅 Or A ↔ ◡𝑅 Or A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  ∃wex 1378   ∈ wcel 1390  ∀wral 2300   class class class wbr 3754   Po wpo 4021   Or wor 4022  ^◡ccnv 4286

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 3865  ax-pow 3917  ax-pr 3934

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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-po 4023  df-iso 4024  df-cnv 4295

This theorem is referenced by:  gtso  6846