Theorem ensn1 9060

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7754. (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 5442	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6837	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6889	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3533	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		snex 5442	. . . 4 ⊢ {𝐴} ∈ V
8		snex 5442	. . . 4 ⊢ {∅} ∈ V
9		breng 8993	. . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
10	7, 8, 9	mp2an 692	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
11	6, 10	mpbir 231	. 2 ⊢ {𝐴} ≈ {∅}
12		df1o2 8512	. 2 ⊢ 1o = {∅}
13	11, 12	breqtrri 5175	1 ⊢ {𝐴} ≈ 1o

Syntax hints:   ↔ wb 206  ∃wex 1776   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  {csn 4631  ⟨cop 4637   class class class wbr 5148  –1-1-onto→wf1o 6562  1_oc1o 8498   ≈ cen 8981

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-1o 8505  df-en 8985

This theorem is referenced by:  ensn1g  9061  en1  9063  sdom1  9276  fodomfiOLD  9368  pm54.43  10039  1nprm  16713  gex1  19624  sylow2a  19652  0frgp  19812  en1top  23007  en2top  23008  t1connperf  23460  ptcmplem2  24077  xrge0tsms2  24871  sconnpi1  35224