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Theorem en2sn 8992
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5325. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7677. (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 5393 . . 3 {⟨𝐴, 𝐵⟩} ∈ V
2 f1osng 6830 . . 3 ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
3 f1oeq1 6777 . . . 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 68 . 2 ((𝐴𝐶𝐵𝐷) → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
6 snex 5393 . . 3 {𝐴} ∈ V
7 snex 5393 . . 3 {𝐵} ∈ V
8 breng 8899 . . 3 (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}))
96, 7, 8mp2an 691 . 2 ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})
105, 9sylibr 233 1 ((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  Vcvv 3448  {csn 4591  cop 4597   class class class wbr 5110  1-1-ontowf1o 6500  cen 8887
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-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-en 8891
This theorem is referenced by:  enrefnn  8998  enpr2dOLD  9001  difsnen  9004  domunsncan  9023  sucdom2OLD  9033  domunsn  9078  limensuci  9104  infensuc  9106  unfi  9123  sucdom2  9157  0sdom1dom  9189  1sdom2dom  9198  dif1ennnALT  9228  dif1card  9953  fin23lem26  10268  unsnen  10496  canthp1lem1  10595  fzennn  13880  hashsng  14276  mreexexlem4d  17534
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