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Theorem opthne 5365
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 5364 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
41, 2, 3mp2an 688 1 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841  wcel 2105  wne 3013  Vcvv 3492  cop 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564
This theorem is referenced by:  m2detleib  21168  addsqnreup  25946  zlmodzxzldeplem  44481  line2x  44669  inlinecirc02plem  44701
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