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Theorem pr2ne 10042
Description: If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5371, ax-un 7754. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
pr2ne ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Proof of Theorem pr2ne
StepHypRef Expression
1 snnen2o 9271 . . . 4 ¬ {𝐴} ≈ 2o
2 dfsn2 4644 . . . . . 6 {𝐴} = {𝐴, 𝐴}
3 preq2 4739 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
42, 3eqtr2id 2788 . . . . 5 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54breq1d 5158 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o))
61, 5mtbiri 327 . . 3 (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o)
76necon2ai 2968 . 2 ({𝐴, 𝐵} ≈ 2o𝐴𝐵)
8 enpr2 10040 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
983expia 1120 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝐵 → {𝐴, 𝐵} ≈ 2o))
107, 9impbid2 226 1 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  {csn 4631  {cpr 4633   class class class wbr 5148  2oc2o 8499  cen 8981
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1o 8505  df-2o 8506  df-en 8985
This theorem is referenced by:  prdom2  10044  isprm2lem  16715  pmtrrn2  19493  mdetunilem7  22640  trsp2cyc  33126  en2pr  43537  pr2cv  43538  pren2  43543
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