Theorem ensn1 8565

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 5322	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6597	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5202	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6647	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3541	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		bren 8510	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
8	6, 7	mpbir 233	. 2 ⊢ {𝐴} ≈ {∅}
9		df1o2 8108	. 2 ⊢ 1o = {∅}
10	8, 9	breqtrri 5084	1 ⊢ {𝐴} ≈ 1o

Syntax hints:  ∃wex 1773   ∈ wcel 2107  Vcvv 3493  ∅c0 4289  {csn 4559  ⟨cop 4565   class class class wbr 5057  –1-1-onto→wf1o 6347  1_oc1o 8087   ≈ cen 8498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-suc 6190  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-1o 8094  df-en 8502

This theorem is referenced by:  ensn1g  8566  en1  8568  fodomfi  8789  pm54.43  9421  1nprm  16015  gex1  18708  sylow2a  18736  0frgp  18897  en1top  21584  en2top  21585  t1connperf  22036  ptcmplem2  22653  xrge0tsms2  23435  sconnpi1  32479