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Theorem en2sn 9081
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5365. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7755. (Revised by BTernaryTau, 25-Sep-2024.)
Assertion
Ref Expression
en2sn ((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})

Proof of Theorem en2sn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 snex 5436 . . 3 {⟨𝐴, 𝐵⟩} ∈ V
2 f1osng 6889 . . 3 ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
3 f1oeq1 6836 . . . 4 (𝑓 = {⟨𝐴, 𝐵⟩} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
43spcegv 3597 . . 3 ({⟨𝐴, 𝐵⟩} ∈ V → ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
51, 2, 4mpsyl 68 . 2 ((𝐴𝐶𝐵𝐷) → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
6 snex 5436 . . 3 {𝐴} ∈ V
7 snex 5436 . . 3 {𝐵} ∈ V
8 breng 8994 . . 3 (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
96, 7, 8mp2an 692 . 2 ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
105, 9sylibr 234 1 ((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1779  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   class class class wbr 5143  1-1-ontowf1o 6560  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-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-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  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-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986
This theorem is referenced by:  enrefnn  9087  enpr2dOLD  9090  difsnen  9093  domunsncan  9112  sucdom2OLD  9122  domunsn  9167  limensuci  9193  infensuc  9195  unfi  9211  sucdom2  9243  0sdom1dom  9274  1sdom2dom  9283  dif1ennnALT  9311  fodomfi  9350  dif1card  10050  fin23lem26  10365  unsnen  10593  canthp1lem1  10692  fzennn  14009  hashsng  14408  mreexexlem4d  17690
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