Theorem ensn1 8285

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 5013	. . . . 5 ⊢ ∅ ∈ V
3	1, 2	f1osn 6416	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4		snex 5128	. . . . 5 ⊢ {⟨𝐴, ∅⟩} ∈ V
5		f1oeq1 6366	. . . . 5 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
6	4, 5	spcev 3516	. . . 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 8230	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
9	7, 8	mpbir 223	. 2 ⊢ {𝐴} ≈ {∅}
10		df1o2 7838	. 2 ⊢ 1o = {∅}
11	9, 10	breqtrri 4899	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1880   ∈ wcel 2166  Vcvv 3413  ∅c0 4143  {csn 4396  ⟨cop 4402   class class class wbr 4872  –1-1-onto→wf1o 6121  1_oc1o 7818   ≈ cen 8218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-suc 5968  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-1o 7825  df-en 8222

This theorem is referenced by:  ensn1g  8286  en1  8288  fodomfi  8507  pm54.43  9138  1nprm  15763  gex1  18356  sylow2a  18384  0frgp  18544  en1top  21158  en2top  21159  t1connperf  21609  ptcmplem2  22226  xrge0tsms2  23007  sconnpi1  31766