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Theorem opthne 5397
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 5396 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
41, 2, 3mp2an 689 1 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844  wcel 2106  wne 2943  Vcvv 3432  cop 4567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568
This theorem is referenced by:  m2detleib  21780  addsqnreup  26591  otthne  33682  xpord2lem  33789  xpord2pred  33792  xpord2ind  33794  zlmodzxzldeplem  45839  line2x  46100  inlinecirc02plem  46132
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