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Theorem opthne 5231
 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⟩ ↔ (𝐴𝐶𝐵𝐷))

Proof of Theorem opthne
StepHypRef Expression
1 opthne.1 . 2 𝐴 ∈ V
2 opthne.2 . 2 𝐵 ∈ V
3 opthneg 5230 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
41, 2, 3mp2an 679 1 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∨ wo 833   ∈ wcel 2050   ≠ wne 2968  Vcvv 3416  ⟨cop 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448 This theorem is referenced by:  m2detleib  20944  addsqnreup  25721  zlmodzxzldeplem  43918  line2x  44107  inlinecirc02plem  44139
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