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Theorem enpr2 10043
Description: An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9090. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5364, ax-un 7756. (Revised by BTernaryTau, 30-Dec-2024.)
Assertion
Ref Expression
enpr2 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)

Proof of Theorem enpr2
StepHypRef Expression
1 df-ne 2940 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 simp1 1136 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐶)
3 simp2 1137 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐷)
4 simp3 1138 . . 3 ((𝐴𝐶𝐵𝐷 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
52, 3, 4enpr2d 9090 . 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 1086   = wceq 1539  wcel 2107  wne 2939  {cpr 4627   class class class wbr 5142  2oc2o 8501  cen 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-suc 6389  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-1o 8507  df-2o 8508  df-en 8987
This theorem is referenced by:  pr2ne  10045  pr2neOLD  10046  en2eqpr  10048  en2eleq  10049  pr2pwpr  14519  pmtrprfv  19472  pmtrprfv3  19473  symggen  19489  pmtr3ncomlem1  19492  pmtr3ncom  19494  mdetralt  22615  en2top  22993  hmphindis  23806  pmtrcnel  33110  pmtrcnel2  33111  fzo0pmtrlast  33113  pmtridf1o  33115  pmtrto1cl  33120  cycpm2tr  33140  cyc3evpm  33171  cyc3genpmlem  33172  cyc3conja  33178
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