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Theorem rrx2plord 48570
Description: The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point iff its first coordinate is less than the first coordinate of the other point, or the first coordinates of both points are equal and the second coordinate of the first point is less than the second coordinate of the other point: 𝑎, 𝑏⟩ ≤ ⟨𝑥, 𝑦 iff (𝑎 < 𝑥 ∨ (𝑎 = 𝑥𝑏𝑦)). (Contributed by AV, 12-Mar-2023.)
Hypothesis
Ref Expression
rrx2plord.o 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
Assertion
Ref Expression
rrx2plord ((𝑋𝑅𝑌𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝑂(𝑥,𝑦)

Proof of Theorem rrx2plord
StepHypRef Expression
1 df-br 5149 . . 3 (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑂)
2 rrx2plord.o . . . 4 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
32eleq2i 2831 . . 3 (⟨𝑋, 𝑌⟩ ∈ 𝑂 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))})
41, 3bitri 275 . 2 (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))})
5 fveq1 6906 . . . . 5 (𝑥 = 𝑋 → (𝑥‘1) = (𝑋‘1))
6 fveq1 6906 . . . . 5 (𝑦 = 𝑌 → (𝑦‘1) = (𝑌‘1))
75, 6breqan12d 5164 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥‘1) < (𝑦‘1) ↔ (𝑋‘1) < (𝑌‘1)))
85, 6eqeqan12d 2749 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥‘1) = (𝑦‘1) ↔ (𝑋‘1) = (𝑌‘1)))
9 fveq1 6906 . . . . . 6 (𝑥 = 𝑋 → (𝑥‘2) = (𝑋‘2))
10 fveq1 6906 . . . . . 6 (𝑦 = 𝑌 → (𝑦‘2) = (𝑌‘2))
119, 10breqan12d 5164 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥‘2) < (𝑦‘2) ↔ (𝑋‘2) < (𝑌‘2)))
128, 11anbi12d 632 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2)) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))
137, 12orbi12d 918 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))) ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
1413opelopab2a 5545 . 2 ((𝑋𝑅𝑌𝑅) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))} ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
154, 14bitrid 283 1 ((𝑋𝑅𝑌𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  {copab 5210  cfv 6563  1c1 11154   < clt 11293  2c2 12319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571
This theorem is referenced by:  rrx2plord1  48571  rrx2plord2  48572  rrx2plordisom  48573
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