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Theorem rrx2plordso 46900
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 11243 . . 3 < Or ℝ
2 eqid 2733 . . . 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 8065 . . 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 691 . 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 2733 . . . 4 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
85, 6, 7, 2rrx2plordisom 46899 . . 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 7297 . . 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 397  wo 846   = wceq 1542  wcel 2107  {cpr 4592  cop 4596   class class class wbr 5109  {copab 5171   Or wor 5548   × cxp 5635  cfv 6500   Isom wiso 6501  (class class class)co 7361  cmpo 7363  1st c1st 7923  2nd c2nd 7924  m cmap 8771  cr 11058  1c1 11060   < clt 11197  2c2 12216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-2 12224
This theorem is referenced by: (None)
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