Theorem elsng 3637

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 3628	. 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 3622

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 3628

This theorem is referenced by:  elsn  3638  elsni  3640  snidg  3651  eltpg  3667  eldifsn  3749  elsucg  4439  funconstss  5680  fniniseg  5682  fniniseg2  5684  tpfidceq  6991  fidcenumlemrks  7019  ltxr  9850  elfzp12  10174  1exp  10660  imasaddfnlemg  12957  0subm  13116  0subg  13329  0nsg  13344  kerf1ghm  13404  lsssn0  13926  plycj  14997  2lgslem2  15333