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Theorem rrx2plordso 46742
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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((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
StepHypRef Expression
1 ltso 11231 . . 3 < Or ℝ
2 eqid 2736 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}
32soxp 8057 . . 3 (( < Or ℝ ∧ < Or ℝ) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} Or (ℝ × ℝ))
41, 1, 3mp2an 690 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} Or (ℝ × ℝ)
5 rrx2plord.o . . . 4 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
6 rrx2plord2.r . . . 4 𝑅 = (ℝ ↑m {1, 2})
7 eqid 2736 . . . 4 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
85, 6, 7, 2rrx2plordisom 46741 . . 3 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅)
9 isoso 7289 . . 3 ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) Isom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅))
108, 9ax-mp 5 . 2 ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅)
114, 10mpbi 229 1 𝑂 Or 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cpr 4586  cop 4590   class class class wbr 5103  {copab 5165   Or wor 5542   × cxp 5629  cfv 6493   Isom wiso 6494  (class class class)co 7353  cmpo 7355  1st c1st 7915  2nd c2nd 7916  m cmap 8761  cr 11046  1c1 11048   < clt 11185  2c2 12204
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-po 5543  df-so 5544  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-2 12212
This theorem is referenced by: (None)
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