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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.
StepHypRef Expression
1 eqeq1 2024 . 2 (x = A → (x = BA = B))
2 df-sn 3352 . 2 {B} = {xx = B}
31, 2elab2g 2662 1 (A 𝑉 → (A {B} ↔ A = B))
Colors of variables: wff set class
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
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