Theorem opth1neg 49636

Description: Two ordered pairs are not equal if their first components are not equal. (Contributed by Zhi Wang, 7-Oct-2025.)

Assertion

Ref	Expression
opth1neg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐶 → ⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩))

Proof of Theorem opth1neg

Step	Hyp	Ref	Expression
1		orc 881	. 2 ⊢ (𝐴 ≠ 𝐶 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
2		opthneg 5466	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
3	1, 2	imbitrrid 249	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐶 → ⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2146   ≠ wne 2961  ⟨cop 4598

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5260  ax-pr 5407

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599

This theorem is used by:  fucofvalne  50135