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Theorem cnvso 6252

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 6251	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ^◡𝑅 Po 𝐴)
2		ralcom 3265	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3		vex 3433	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5837	. . . . . 6 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2020	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5837	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1157	. . . . 5 ⊢ ((𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	8	2ralbii 3112	. . . 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 5540	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5540	. 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 3051 class class class wbr 5085 Po wpo 5537 Or wor 5538 ^◡ccnv 5630

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 2708 ax-sep 5231 ax-pr 5375

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 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-br 5086 df-opab 5148 df-po 5539 df-so 5540 df-cnv 5639

This theorem is referenced by: infexd 9397 eqinf 9398 infval 9400 infcl 9402 inflb 9403 infglb 9404 infglbb 9405 fiinfcl 9416 infltoreq 9417 infempty 9422 infiso 9423 wofib 9460 oemapso 9603 cflim2 10185 fin23lem40 10273 gtso 11227 nomaxmo 27662 tosglb 33035 xrsclat 33071 xrge0iifiso 34079 socnv 35946 welb 38057 xrgtso 45775

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