Theorem elsncg 3368

Description: There is only 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
elsncg	⊢ (A ∈ 𝑉 → (A ∈ {B} ↔ A = B))

Proof of Theorem elsncg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2024	. 2 ⊢ (x = A → (x = B ↔ A = B))
2		df-sn 3352	. 2 ⊢ {B} = {x ∣ x = B}
3	1, 2	elab2g 2662	1 ⊢ (A ∈ 𝑉 → (A ∈ {B} ↔ A = B))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370  {csn 3346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sn 3352

This theorem is referenced by:  elsnc  3369  elsni  3370  snidg  3371  eltpg  3386  eldifsn  3465  elsucg  4086  funconstss  5206  fniniseg  5208  fniniseg2  5210