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Theorem opthne 5484
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 5483 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
41, 2, 3mp2an 691 1 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 846  wcel 2099  wne 2937  Vcvv 3471  cop 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636
This theorem is referenced by:  xpord2lem  8147  xpord2pred  8150  xpord2indlem  8152  m2detleib  22546  addsqnreup  27389  mulsval  28022  zlmodzxzldeplem  47566  line2x  47827  inlinecirc02plem  47859
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