Theorem ensn1OLD 9083

Description: Obsolete version of ensn1 9082 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 5451	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6850	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1OLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6902	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3545	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 9013	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 231	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8529	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5193	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  {csn 4648  ⟨cop 4654   class class class wbr 5166  –1-1-onto→wf1o 6572  1_oc1o 8515   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-suc 6401  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-1o 8522  df-en 9004

This theorem is referenced by: (None)