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Theorem pr2ne 10044
Description: If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5365, ax-un 7755. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
pr2ne ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Proof of Theorem pr2ne
StepHypRef Expression
1 snnen2o 9273 . . . 4 ¬ {𝐴} ≈ 2o
2 dfsn2 4639 . . . . . 6 {𝐴} = {𝐴, 𝐴}
3 preq2 4734 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
42, 3eqtr2id 2790 . . . . 5 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54breq1d 5153 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o))
61, 5mtbiri 327 . . 3 (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o)
76necon2ai 2970 . 2 ({𝐴, 𝐵} ≈ 2o𝐴𝐵)
8 enpr2 10042 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
983expia 1122 . 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 1540  wcel 2108  wne 2940  {csn 4626  {cpr 4628   class class class wbr 5143  2oc2o 8500  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-2o 8507  df-en 8986
This theorem is referenced by:  prdom2  10046  isprm2lem  16718  pmtrrn2  19478  mdetunilem7  22624  trsp2cyc  33143  en2pr  43560  pr2cv  43561  pren2  43566
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