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Theorem ensn1OLD 9014
Description: Obsolete version of ensn1 9013 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 5430 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6818 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1OLD.1 . . . . 5 𝐴 ∈ V
4 0ex 5306 . . . . 5 ∅ ∈ V
53, 4f1osn 6870 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3528 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 bren 8945 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
86, 7mpbir 230 . 2 {𝐴} ≈ {∅}
9 df1o2 8469 . 2 1o = {∅}
108, 9breqtrri 5174 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wex 1781  wcel 2106  Vcvv 3474  c0 4321  {csn 4627  cop 4633   class class class wbr 5147  1-1-ontowf1o 6539  1oc1o 8455  cen 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-suc 6367  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-1o 8462  df-en 8936
This theorem is referenced by: (None)
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