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Theorem opthne 5488
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 5487 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
41, 2, 3mp2an 690 1 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845  wcel 2099  wne 2930  Vcvv 3462  cop 4639
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640
This theorem is referenced by:  xpord2lem  8156  xpord2pred  8159  xpord2indlem  8161  m2detleib  22624  addsqnreup  27472  mulsval  28110  zlmodzxzldeplem  47881  line2x  48142  inlinecirc02plem  48174
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