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Theorem mosn 3447
Description: A singleton has at most one element. This works whether  A is a proper class or not, and in that sense can be seen as encompassing both snmg 3526 and snprc 3475. (Contributed by Jim Kingdon, 30-Aug-2018.)
Assertion
Ref Expression
mosn  |-  E* x  x  e.  { A }
Distinct variable group:    x, A

Proof of Theorem mosn
StepHypRef Expression
1 moeq 2776 . 2  |-  E* x  x  =  A
2 velsn 3433 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
32mobii 1980 . 2  |-  ( E* x  x  e.  { A }  <->  E* x  x  =  A )
41, 3mpbir 144 1  |-  E* x  x  e.  { A }
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   E*wmo 1944   {csn 3416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sn 3422
This theorem is referenced by: (None)
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