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Theorem prneprprc 4610
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 4609 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐷})
21adantr 474 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐷})
3 prprc1 4518 . . . 4 𝐶 ∈ V → {𝐶, 𝐷} = {𝐷})
43neeq2d 3059 . . 3 𝐶 ∈ V → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
54adantl 475 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
62, 5mpbird 249 1 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1113  wcel 2166  wne 2999  Vcvv 3414  {csn 4397  {cpr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-un 3803  df-nul 4145  df-sn 4398  df-pr 4400
This theorem is referenced by:  preq12nebg  4613  opthprneg  4615
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