Theorem opthne 5442

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

Hypotheses

Ref	Expression
opthne.1	⊢ 𝐴 ∈ V
opthne.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opthne	⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Proof of Theorem opthne

Step	Hyp	Ref	Expression
1		opthne.1	. 2 ⊢ 𝐴 ∈ V
2		opthne.2	. 2 ⊢ 𝐵 ∈ V
3		opthneg 5441	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
4	1, 2, 3	mp2an 692	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ⟨cop 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596

This theorem is referenced by:  xpord2lem  8121  xpord2pred  8124  xpord2indlem  8126  m2detleib  22518  addsqnreup  27354  mulsval  28012  gpgedg2ov  48054  gpgedg2iv  48055  gpg5nbgrvtx03starlem1  48056  gpg5nbgrvtx03starlem2  48057  gpg5nbgrvtx03starlem3  48058  gpg5nbgrvtx13starlem1  48059  gpg5nbgrvtx13starlem2  48060  gpg5nbgrvtx13starlem3  48061  gpg3nbgrvtx0  48064  gpg3nbgrvtx0ALT  48065  gpg3nbgrvtx1  48066  gpg3kgrtriex  48077  gpgprismgr4cycllem2  48083  gpgprismgr4cycllem7  48088  zlmodzxzldeplem  48484  line2x  48740  inlinecirc02plem  48772