Theorem opthneg 5449

Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)

Assertion

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

Proof of Theorem opthneg

Step	Hyp	Ref	Expression
1		df-ne 2958	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ ¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
2		opthg 5445	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
3	2	notbid 320	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4		ianor 995	. . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5		df-ne 2958	. . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
6		df-ne 2958	. . . . 5 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷)
7	5, 6	orbi12i 925	. . . 4 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
8	4, 7	bitr4i 280	. . 3 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
9	3, 8	bitrdi 289	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
10	1, 9	bitrid 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  opthne  5450  addsqnreup  27507  addsval  28055  mulsval  28202  linds2eq  33567  gpgusgralem  48678  gpg5nbgrvtx03starlem1  48690  gpg5nbgrvtx03starlem3  48692  gpg5nbgrvtx13starlem1  48693  gpg5nbgrvtx13starlem3  48695  zlmodzxznm  49119  opth1neg  49447  opth2neg  49448