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Theorem cnvso 6246

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 𝐴 ↔ ^◡𝑅 Or 𝐴)

Proof of Theorem cnvso Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvpo 6245	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ^◡𝑅 Po 𝐴)
2		ralcom 3266	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5831	. . . . . 6 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2020	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5831	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1157	. . . . 5 ⊢ ((𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	8	2ralbii 3113	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
10	2, 9	bitr4i 278	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦))
11	1, 10	anbi12i 629	. 2 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (^◡𝑅 Po 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦)))
12		df-so 5533	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5533	. 2 ⊢ (^◡𝑅 Or 𝐴 ↔ (^◡𝑅 Po 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦)))
14	11, 12, 13	3bitr4i 303	1 ⊢ (𝑅 Or 𝐴 ↔ ^◡𝑅 Or 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∨ w3o 1086 ∀wral 3052 class class class wbr 5086 Po wpo 5530 Or wor 5531 ^◡ccnv 5623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-br 5087 df-opab 5149 df-po 5532 df-so 5533 df-cnv 5632

This theorem is referenced by: infexd 9390 eqinf 9391 infval 9393 infcl 9395 inflb 9396 infglb 9397 infglbb 9398 fiinfcl 9409 infltoreq 9410 infempty 9415 infiso 9416 wofib 9453 oemapso 9594 cflim2 10176 fin23lem40 10264 gtso 11218 nomaxmo 27676 tosglb 33050 xrsclat 33086 xrge0iifiso 34095 socnv 35962 welb 38071 xrgtso 45793

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