Theorem opthneg 5465

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 2961	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ ¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
2		opthg 5461	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
3	2	notbid 321	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4		ianor 997	. . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5		df-ne 2961	. . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
6		df-ne 2961	. . . . 5 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷)
7	5, 6	orbi12i 928	. . . 4 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
8	4, 7	bitr4i 281	. . 3 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
9	3, 8	bitrdi 290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
10	1, 9	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ⟨cop 4597

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 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is used by:  opthne  5466  addsqnreup  27660  addsval  28208  mulsval  28355  linds2eq  33760  gpgusgralem  48881  gpg5nbgrvtx03starlem1  48893  gpg5nbgrvtx03starlem3  48895  gpg5nbgrvtx13starlem1  48896  gpg5nbgrvtx13starlem3  48898  zlmodzxznm  49336  opth1neg  49663  opth2neg  49664