Theorem ensn1 8567

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		snex 5323	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6598	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5203	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6648	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3542	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8512	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 233	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8110	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5085	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4560  ⟨cop 4566   class class class wbr 5058  –1-1-onto→wf1o 6348  1_oc1o 8089   ≈ cen 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-suc 6191  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-1o 8096  df-en 8504

This theorem is referenced by:  ensn1g  8568  en1  8570  fodomfi  8791  pm54.43  9423  1nprm  16017  gex1  18710  sylow2a  18738  0frgp  18899  en1top  21586  en2top  21587  t1connperf  22038  ptcmplem2  22655  xrge0tsms2  23437  sconnpi1  32481