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Theorem ensn1OLD 8965
Description: Obsolete version of ensn1 8964 as of 23-Sep-2024. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ensn1OLD.1 𝐴 ∈ V
Assertion
Ref Expression
ensn1OLD {𝐴} ≈ 1o

Proof of Theorem ensn1OLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 snex 5389 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6773 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1OLD.1 . . . . 5 𝐴 ∈ V
4 0ex 5265 . . . . 5 ∅ ∈ V
53, 4f1osn 6825 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3496 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 bren 8896 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
86, 7mpbir 230 . 2 {𝐴} ≈ {∅}
9 df1o2 8420 . 2 1o = {∅}
108, 9breqtrri 5133 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wex 1782  wcel 2107  Vcvv 3444  c0 4283  {csn 4587  cop 4593   class class class wbr 5106  1-1-ontowf1o 6496  1oc1o 8406  cen 8883
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-1o 8413  df-en 8887
This theorem is referenced by: (None)
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