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Theorem mosn 3530
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 3611 and snprc 3558. (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 2832 . 2 ∃*𝑥 𝑥 = 𝐴
2 velsn 3514 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32mobii 2014 . 2 (∃*𝑥 𝑥 ∈ {𝐴} ↔ ∃*𝑥 𝑥 = 𝐴)
41, 3mpbir 145 1 ∃*𝑥 𝑥 ∈ {𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  ∃*wmo 1978  {csn 3497
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sn 3503
This theorem is referenced by: (None)
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