Theorem ensn1OLD 8808

Description: Obsolete version of ensn1 8807 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.

Step	Hyp	Ref	Expression
1		snex 5354	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6704	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1OLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5231	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6756	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3481	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8743	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 230	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8304	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5101	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074  –1-1-onto→wf1o 6432  1_oc1o 8290   ≈ cen 8730

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-1o 8297  df-en 8734

This theorem is referenced by: (None)