cnvso - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnvso		Structured version Visualization version GIF version

Theorem cnvso 6277

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 6276	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ^◡𝑅 Po 𝐴)
2		ralcom 3270	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3		vex 3463	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5862	. . . . . 6 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2017	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5862	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1156	. . . . 5 ⊢ ((𝑦^◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	8	2ralbii 3115	. . . 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 628	. 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 5562	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5562	. 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 1085 ∀wral 3051 class class class wbr 5119 Po wpo 5559 Or wor 5560 ^◡ccnv 5653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ne 2933 df-ral 3052 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-br 5120 df-opab 5182 df-po 5561 df-so 5562 df-cnv 5662

This theorem is referenced by: infexd 9496 eqinf 9497 infval 9499 infcl 9501 inflb 9502 infglb 9503 infglbb 9504 fiinfcl 9515 infltoreq 9516 infempty 9521 infiso 9522 wofib 9559 oemapso 9696 cflim2 10277 fin23lem40 10365 gtso 11316 nomaxmo 27662 tosglb 32955 xrsclat 33003 xrge0iifiso 33966 socnv 35781 welb 37760 xrgtso 45372

W3C validator