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Theorem prneprprc 4524
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 4523 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐷})
21adantr 466 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐷})
3 prprc1 4434 . . . 4 𝐶 ∈ V → {𝐶, 𝐷} = {𝐷})
43neeq2d 3002 . . 3 𝐶 ∈ V → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
54adantl 467 . 2 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷}))
62, 5mpbird 247 1 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070  wcel 2144  wne 2942  Vcvv 3349  {csn 4314  {cpr 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-v 3351  df-dif 3724  df-un 3726  df-nul 4062  df-sn 4315  df-pr 4317
This theorem is referenced by:  preq12nebg  4527  opthprneg  4529
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