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Theorem prneprprc 4790
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 4789 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐷})
21adantr 483 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐷})
3 prprc1 4700 . . . 4 𝐶 ∈ V → {𝐶, 𝐷} = {𝐷})
43neeq2d 3076 . . 3 𝐶 ∈ V → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
54adantl 484 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
62, 5mpbird 259 1 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083  wcel 2110  wne 3016  Vcvv 3494  {csn 4566  {cpr 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4567  df-pr 4569
This theorem is referenced by:  preq12nebg  4792  opthprneg  4794
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