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Theorem enpr2 9997
Description: An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9049. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5364, ax-un 7725. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
enpr2 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)

Proof of Theorem enpr2
StepHypRef Expression
1 df-ne 2942 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 simp1 1137 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐶)
3 simp2 1138 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐷)
4 simp3 1139 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
52, 3, 4enpr2d 9049 . 2 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ≈ 2o)
61, 5syl3an3b 1406 1 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2941  {cpr 4631   class class class wbr 5149  2oc2o 8460  cen 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-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-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-1o 8466  df-2o 8467  df-en 8940
This theorem is referenced by:  pr2ne  9999  pr2neOLD  10000  en2eqpr  10002  en2eleq  10003  pr2pwpr  14440  pmtrprfv  19321  pmtrprfv3  19322  symggen  19338  pmtr3ncomlem1  19341  pmtr3ncom  19343  mdetralt  22110  en2top  22488  hmphindis  23301  pmtrcnel  32250  pmtrcnel2  32251  pmtridf1o  32253  pmtrto1cl  32258  cycpm2tr  32278  cyc3evpm  32309  cyc3genpmlem  32310  cyc3conja  32316
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