Theorem elsn 3682

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elsn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elsn	⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Proof of Theorem elsn

Step	Hyp	Ref	Expression
1		elsn.1	. 2 ⊢ 𝐴 ∈ V
2		elsng 3681	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799  {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  velsn  3683  sneqr  3837  onsucelsucexmid  4621  ordsoexmid  4653  opthprc  4769  dmsnm  5193  dmsnopg  5199  cnvcnvsn  5204  sniota  5308  fsn  5806  eusvobj2  5986  mapdm0  6808  djulclb  7218  pw1nel3  7412  sucpw1nel3  7414  opelreal  8010