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Theorem pr2ne 10003
Description: If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5364, ax-un 7729. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
pr2ne ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Proof of Theorem pr2ne
StepHypRef Expression
1 snnen2o 9241 . . . 4 ¬ {𝐴} ≈ 2o
2 dfsn2 4642 . . . . . 6 {𝐴} = {𝐴, 𝐴}
3 preq2 4739 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
42, 3eqtr2id 2783 . . . . 5 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
54breq1d 5159 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o))
61, 5mtbiri 326 . . 3 (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o)
76necon2ai 2968 . 2 ({𝐴, 𝐵} ≈ 2o𝐴𝐵)
8 enpr2 10001 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
983expia 1119 . 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 394   = wceq 1539  wcel 2104  wne 2938  {csn 4629  {cpr 4631   class class class wbr 5149  2oc2o 8464  cen 8940
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8470  df-2o 8471  df-en 8944
This theorem is referenced by:  prdom2  10005  isprm2lem  16624  pmtrrn2  19371  mdetunilem7  22342  trsp2cyc  32550  en2pr  42602  pr2cv  42603  pren2  42608
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