Theorem ensn1 8953

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7675. (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 5378	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6756	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5249	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6808	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3490	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		snex 5378	. . . 4 ⊢ {𝐴} ∈ V
8		snex 5378	. . . 4 ⊢ {∅} ∈ V
9		breng 8888	. . . 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 8402	. 2 ⊢ 1o = {∅}
13	11, 12	breqtrri 5122	1 ⊢ {𝐴} ≈ 1o

Syntax hints:   ↔ wb 206  ∃wex 1779   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  {csn 4579  ⟨cop 4585   class class class wbr 5095  –1-1-onto→wf1o 6485  1_oc1o 8388   ≈ cen 8876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-suc 6317  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-1o 8395  df-en 8880

This theorem is referenced by:  ensn1g  8954  en1  8956  sdom1  9149  fodomfiOLD  9239  pm54.43  9916  1nprm  16609  gex1  19489  sylow2a  19517  0frgp  19677  en1top  22888  en2top  22889  t1connperf  23340  ptcmplem2  23957  xrge0tsms2  24741  sconnpi1  35231  setcsnterm  49495