rrx2plordso - Mathbox for Alexander van der Vekens

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Theorem rrx2plordso 48704

Description: The lexicographical ordering for points in the two dimensional Euclidean plane is a strict total ordering. (Contributed by AV, 12-Mar-2023.)

**Hypotheses**
Ref	Expression
rrx2plord.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
rrx2plord2.r	⊢ 𝑅 = (ℝ ↑_m {1, 2})

**Assertion**
Ref	Expression
rrx2plordso	⊢ 𝑂 Or 𝑅

Distinct variable group: 𝑥,𝑅,𝑦 Allowed substitution hints: 𝑂(𝑥,𝑦)

**Proof of Theorem rrx2plordso**
Step	Hyp	Ref	Expression
1		ltso 11315	. . 3 ⊢ < Or ℝ
2		eqid 2735	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))}
3	2	soxp 8128	. . 3 ⊢ (( < Or ℝ ∧ < Or ℝ) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))} Or (ℝ × ℝ))
4	1, 1, 3	mp2an 692	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))} Or (ℝ × ℝ)
5		rrx2plord.o	. . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
6		rrx2plord2.r	. . . 4 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
7		eqid 2735	. . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
8	5, 6, 7, 2	rrx2plordisom 48703	. . 3 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)
9		isoso 7341	. . 3 ⊢ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅))
10	8, 9	ax-mp 5	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1^st ‘𝑥) < (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) < (2^nd ‘𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅)
11	4, 10	mpbi 230	1 ⊢ 𝑂 Or 𝑅

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 {cpr 4603 ⟨cop 4607 class class class wbr 5119 {copab 5181 Or wor 5560 × cxp 5652 ‘cfv 6531 Isom wiso 6532 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7986 2^nd c2nd 7987 ↑_m cmap 8840 ℝcr 11128 1c1 11130 < clt 11269 2c2 12295

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-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-2 12303

This theorem is referenced by: (None)

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