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Theorem elsni 3374
Description: There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
elsni (A {B} → A = B)

Proof of Theorem elsni
StepHypRef Expression
1 elsncg 3372 . 2 (A {B} → (A {B} ↔ A = B))
21ibi 165 1 (A {B} → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  {csn 3350
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sn 3356
This theorem is referenced by:  elsnc2g  3379  disjsn2  3407  sssnm  3499  disjxsn  3736  opth1  3947  elsuci  4089  ordtri2orexmid  4195  onsucsssucexmid  4196  sosng  4340  ressn  4785  funcnvsn  4871  fvconst  5276  fmptap  5278  fmptapd  5279  fvunsng  5282  1stconst  5765  2ndconst  5766  reldmtpos  5790  tpostpos  5801  elreal2  6542  ax1rid  6571  ltxrlt  6686  bj-nntrans  7320  bj-nnelirr  7322
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