Theorem elsng 3638

Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
elsng	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsng

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2203	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
2		df-sn 3629	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
3	1, 2	elab2g 2911	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {csn 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3629

This theorem is referenced by:  elsn  3639  elsni  3641  snidg  3652  eltpg  3668  eldifsn  3750  elsucg  4440  funconstss  5683  fniniseg  5685  fniniseg2  5687  tpfidceq  7000  fidcenumlemrks  7028  ltxr  9867  elfzp12  10191  1exp  10677  imasaddfnlemg  13016  0subm  13186  0subg  13405  0nsg  13420  kerf1ghm  13480  lsssn0  14002  plycj  15081  2lgslem2  15417