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Theorem mosn 3398
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 3477 and snprc 3426. (Contributed by Jim Kingdon, 30-Aug-2018.)
Assertion
Ref Expression
mosn ∃*x x {A}
Distinct variable group:   x,A

Proof of Theorem mosn
StepHypRef Expression
1 moeq 2710 . 2 ∃*x x = A
2 elsn 3382 . . 3 (x {A} ↔ x = A)
32mobii 1934 . 2 (∃*x x {A} ↔ ∃*x x = A)
41, 3mpbir 134 1 ∃*x x {A}
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  ∃*wmo 1898  {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-sn 3373
This theorem is referenced by: (None)
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