Theorem ensn1 9082

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7770. (Revised by BTernaryTau, 23-Sep-2024.)

Hypothesis

Ref	Expression
ensn1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ensn1	⊢ {𝐴} ≈ 1o

Proof of Theorem ensn1

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		ensn1.1	. . . . 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		snex 5451	. . . 4 ⊢ {𝐴} ∈ V
8		snex 5451	. . . 4 ⊢ {∅} ∈ V
9		breng 9012	. . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
10	7, 8, 9	mp2an 691	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
11	6, 10	mpbir 231	. 2 ⊢ {𝐴} ≈ {∅}
12		df1o2 8529	. 2 ⊢ 1o = {∅}
13	11, 12	breqtrri 5193	1 ⊢ {𝐴} ≈ 1o

Syntax hints:   ↔ wb 206  ∃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

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-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  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:  ensn1g  9084  en1  9086  en1OLD  9087  sdom1  9305  fodomfiOLD  9398  pm54.43  10070  1nprm  16726  gex1  19633  sylow2a  19661  0frgp  19821  en1top  23012  en2top  23013  t1connperf  23465  ptcmplem2  24082  xrge0tsms2  24876  sconnpi1  35207