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Theorem ensn1OLD 9020
Description: Obsolete version of ensn1 9019 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 5424 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6815 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1OLD.1 . . . . 5 𝐴 ∈ V
4 0ex 5300 . . . . 5 ∅ ∈ V
53, 4f1osn 6867 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3523 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 bren 8951 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
86, 7mpbir 230 . 2 {𝐴} ≈ {∅}
9 df1o2 8474 . 2 1o = {∅}
108, 9breqtrri 5168 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wex 1773  wcel 2098  Vcvv 3468  c0 4317  {csn 4623  cop 4629   class class class wbr 5141  1-1-ontowf1o 6536  1oc1o 8460  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-suc 6364  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-1o 8467  df-en 8942
This theorem is referenced by: (None)
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