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Theorem elsn2g 3404
 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
elsn2g

Proof of Theorem elsn2g
StepHypRef Expression
1 elsni 3393 . 2
2 snidg 3400 . . 3
3 eleq1 2100 . . 3
42, 3syl5ibrcom 146 . 2
51, 4impbid2 131 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wceq 1243   wcel 1393  csn 3375 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381 This theorem is referenced by:  elsn2  3405  elsuc2g  4142  mptiniseg  4815  elfzp1  8934  fzosplitsni  9091  iseqid3  9245
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