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Theorem snmg 3650
 Description: The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
snmg (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem snmg
StepHypRef Expression
1 snidg 3562 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 elex2 2706 . 2 (𝐴 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴})
31, 2syl 14 1 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wex 1469   ∈ wcel 1481  {csn 3533 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-sn 3539 This theorem is referenced by:  snm  3652  prmg  3653  exmidsssnc  4136  xpimasn  4998  1stconst  6129  2ndconst  6130
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