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Theorem prneprprc 4869
Description: A proper unordered pair is not an improper unordered pair. (Contributed by AV, 13-Jun-2022.)
Assertion
Ref Expression
prneprprc (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})

Proof of Theorem prneprprc
StepHypRef Expression
1 prnesn 4868 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐷})
21adantr 479 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐷})
3 prprc1 4774 . . . 4 𝐶 ∈ V → {𝐶, 𝐷} = {𝐷})
43neeq2d 2991 . . 3 𝐶 ∈ V → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
54adantl 480 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
62, 5mpbird 256 1 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084  wcel 2099  wne 2930  Vcvv 3462  {csn 4633  {cpr 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 2697
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-v 3464  df-dif 3950  df-un 3952  df-nul 4326  df-sn 4634  df-pr 4636
This theorem is referenced by:  preq12nebg  4871  opthprneg  4873
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