Theorem ensn1 8020

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)

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		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
2		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
3	1, 2	f1osn 6176	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4		snex 4908	. . . . 5 ⊢ {⟨𝐴, ∅⟩} ∈ V
5		f1oeq1 6127	. . . . 5 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
6	4, 5	spcev 3300	. . . 4 ⊢ ({⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
7	3, 6	ax-mp 5	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
8		bren 7964	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
9	7, 8	mpbir 221	. 2 ⊢ {𝐴} ≈ {∅}
10		df1o2 7572	. 2 ⊢ 1𝑜 = {∅}
11	9, 10	breqtrri 4680	1 ⊢ {𝐴} ≈ 1𝑜

Syntax hints:  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653  –1-1-onto→wf1o 5887  1_𝑜c1o 7553   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-en 7956

This theorem is referenced by:  ensn1g  8021  en1  8023  fodomfi  8239  pm54.43  8826  1nprm  15392  gex1  18006  sylow2a  18034  0frgp  18192  en1top  20788  en2top  20789  t1connperf  21239  ptcmplem2  21857  xrge0tsms2  22638  sconnpi1  31221