Theorem rrx2plord1 50037

Description: The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point if its first coordinate is less than the first coordinate of the other point. (Contributed by AV, 12-Mar-2023.)

Hypothesis

Ref	Expression
rrx2plord.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))}

Assertion

Ref	Expression
rrx2plord1	⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘0) < (𝑌‘0)) → 𝑋𝑂𝑌)

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝑂(𝑥, 𝑦)

Proof of Theorem rrx2plord1

Step	Hyp	Ref	Expression
1		simp3 1156	. . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘0) < (𝑌‘0)) → (𝑋‘0) < (𝑌‘0))
2	1	orcd 887	. 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘0) < (𝑌‘0)) → ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1))))
3		rrx2plord.o	. . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))}
4	3	rrx2plord 50036	. . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))
5	4	3adant3 1150	. 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘0) < (𝑌‘0)) → (𝑋𝑂𝑌 ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))
6	2, 5	mpbird 260	1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘0) < (𝑌‘0)) → 𝑋𝑂𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  {copab 5170  ‘cfv 6555  0cc0 11224  1c1 11225   < clt 11367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-iota 6509  df-fv 6563

This theorem is used by: (None)