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Theorem n0mmoeu 3214
Description: A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)
Assertion
Ref Expression
n0mmoeu (x x A → (∃*x x A∃!x x A))
Distinct variable group:   x,A

Proof of Theorem n0mmoeu
StepHypRef Expression
1 exmoeu2 1930 1 (x x A → (∃*x x A∃!x x A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wex 1362   wcel 1374  ∃!weu 1882  ∃*wmo 1883
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
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886
This theorem is referenced by: (None)
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