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Theorem cnvso 6131

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 6130	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ^◡𝑅 Po 𝐴)
2		ralcom 3257	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3		vex 3402	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3402	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5736	. . . . . 6 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2028	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5736	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1158	. . . . 5 ⊢ ((𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	8	2ralbii 3079	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
10	2, 9	bitr4i 281	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦))
11	1, 10	anbi12i 630	. 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 5454	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5454	. 2 ⊢ (^◡𝑅 Or 𝐴 ↔ (^◡𝑅 Po 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦)))
14	11, 12, 13	3bitr4i 306	1 ⊢ (𝑅 Or 𝐴 ↔ ^◡𝑅 Or 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 ∨ w3o 1088 ∀wral 3051 class class class wbr 5039 Po wpo 5451 Or wor 5452 ^◡ccnv 5535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-po 5453 df-so 5454 df-cnv 5544

This theorem is referenced by: infexd 9077 eqinf 9078 infval 9080 infcl 9082 inflb 9083 infglb 9084 infglbb 9085 fiinfcl 9095 infltoreq 9096 infempty 9101 infiso 9102 wofib 9139 oemapso 9275 cflim2 9842 fin23lem40 9930 gtso 10879 tosglb 30926 xrsclat 30962 xrge0iifiso 31553 socnv 33401 nomaxmo 33587 welb 35580 xrgtso 42498

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