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Theorem mosn 3609
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 3691 and snprc 3638. (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 2899 . 2  |-  E* x  x  =  A
2 velsn 3590 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
32mobii 2050 . 2  |-  ( E* x  x  e.  { A }  <->  E* x  x  =  A )
41, 3mpbir 145 1  |-  E* x  x  e.  { A }
Colors of variables: wff set class
Syntax hints:    = wceq 1342   E*wmo 2014    e. wcel 2135   {csn 3573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-sn 3579
This theorem is referenced by: (None)
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