Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsnc2g Unicode version

Theorem elsnc2g 3642
 Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsnc2g

Proof of Theorem elsnc2g
StepHypRef Expression
1 elsni 3638 . 2
2 snidg 3639 . . 3
3 eleq1 2318 . . 3
42, 3syl5ibrcom 215 . 2
51, 4impbid2 197 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wceq 1619   wcel 1621  csn 3614 This theorem is referenced by:  elsnc2  3643  elsuc2g  4432  mptiniseg  5154  fzosplitsni  10888  limcco  19206  ply1termlem  19548  stirlinglem8  27199  elpmapat  29203 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-sn 3620
 Copyright terms: Public domain W3C validator