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Theorem snmg 3513
 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 3427 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 elex2 2587 . 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 1397   ∈ wcel 1409  {csn 3402 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3408 This theorem is referenced by:  snm  3515  prmg  3516  xpimasn  4796  1stconst  5869  2ndconst  5870
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