Theorem ensn1OLD 8762

Description: Obsolete version of ensn1 8761 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 5349	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6688	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1OLD.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6739	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3471	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8701	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 230	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8279	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5097	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  {csn 4558  ⟨cop 4564   class class class wbr 5070  –1-1-onto→wf1o 6417  1_oc1o 8260   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-suc 6257  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-1o 8267  df-en 8692

This theorem is referenced by: (None)