Theorem ensn1OLD 9051

Description: Obsolete version of ensn1 9050 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 5437	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6832	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1OLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5311	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6884	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3528	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8982	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 230	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8502	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5179	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1773   ∈ wcel 2098  Vcvv 3473  ∅c0 4326  {csn 4632  ⟨cop 4638   class class class wbr 5152  –1-1-onto→wf1o 6552  1_oc1o 8488   ≈ cen 8969

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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-suc 6380  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-1o 8495  df-en 8973

This theorem is referenced by: (None)