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Theorem en2sn 9026
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5326. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7722. (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 5400 . . 3 {⟨𝐴, 𝐵⟩} ∈ V
2 f1osng 6853 . . 3 ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
3 f1oeq1 6798 . . . 4 (𝑓 = {⟨𝐴, 𝐵⟩} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
43spcegv 3559 . . 3 ({⟨𝐴, 𝐵⟩} ∈ V → ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
51, 2, 4mpsyl 69 . 2 ((𝐴𝐶𝐵𝐷) → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
6 snex 5400 . . 3 {𝐴} ∈ V
7 snex 5400 . . 3 {𝐵} ∈ V
8 breng 8940 . . 3 (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
96, 7, 8mp2an 704 . 2 ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
105, 9sylibr 237 1 ((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 400  wex 1802  wcel 2145  Vcvv 3457  {csn 4585  cop 4591   class class class wbr 5104  1-1-ontowf1o 6524  cen 8928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-en 8932
This theorem is referenced by:  enrefnn  9031  difsnen  9035  domunsncan  9053  domunsn  9103  limensuci  9129  infensuc  9131  unfi  9143  sucdom2  9175  0sdom1dom  9194  1sdom2dom  9202  dif1ennnALT  9225  fodomfi  9260  dif1card  9982  fin23lem26  10297  unsnen  10525  canthp1lem1  10625  fzennn  13992  hashsng  14393  mreexexlem4d  17691
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