Theorem elsncg 3389

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 2043	. 2 ⊢ (x = A → (x = B ↔ A = B))
2		df-sn 3373	. 2 ⊢ {B} = {x ∣ x = B}
3	1, 2	elab2g 2683	1 ⊢ (A ∈ 𝑉 → (A ∈ {B} ↔ A = B))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373

This theorem is referenced by:  elsnc  3390  elsni  3391  snidg  3392  eltpg  3407  eldifsn  3486  elsucg  4107  funconstss  5228  fniniseg  5230  fniniseg2  5232  ltxr  8465  elfzp12  8731  1exp  8938