Theorem rrx2plord1 49755

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‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}

Assertion

Ref	Expression
rrx2plord1	⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 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‘1) < (𝑌‘1)) → (𝑋‘1) < (𝑌‘1))
2	1	orcd 887	. 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))
3		rrx2plord.o	. . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
4	3	rrx2plord 49754	. . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
5	4	3adant3 1150	. 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
6	2, 5	mpbird 260	1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5102  {copab 5166  ‘cfv 6527  1c1 11172   < clt 11314  2c2 12366

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6483  df-fv 6535

This theorem is used by: (None)