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Theorem pr2ne 9995
Description: If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5362, ax-un 7721. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
pr2ne ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Proof of Theorem pr2ne
StepHypRef Expression
1 snnen2o 9233 . . . 4 ¬ {𝐴} ≈ 2o
2 dfsn2 4640 . . . . . 6 {𝐴} = {𝐴, 𝐴}
3 preq2 4737 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
42, 3eqtr2id 2785 . . . . 5 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54breq1d 5157 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o))
61, 5mtbiri 326 . . 3 (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o)
76necon2ai 2970 . 2 ({𝐴, 𝐵} ≈ 2o𝐴𝐵)
8 enpr2 9993 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
983expia 1121 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝐵 → {𝐴, 𝐵} ≈ 2o))
107, 9impbid2 225 1 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  {csn 4627  {cpr 4629   class class class wbr 5147  2oc2o 8456  cen 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8462  df-2o 8463  df-en 8936
This theorem is referenced by:  prdom2  9997  isprm2lem  16614  pmtrrn2  19322  mdetunilem7  22111  trsp2cyc  32269  en2pr  42283  pr2cv  42284  pren2  42289
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