Theorem ensn1OLD 8673

Description: Obsolete version of ensn1 8672 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 5309	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6627	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1OLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5185	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6678	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3447	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8614	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 234	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8192	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5066	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1787   ∈ wcel 2112  Vcvv 3398  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039  –1-1-onto→wf1o 6357  1_oc1o 8173   ≈ cen 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-suc 6197  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-1o 8180  df-en 8605

This theorem is referenced by: (None)