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Theorem snssg 3626
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
Assertion
Ref Expression
snssg (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))

Proof of Theorem snssg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2180 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 sneq 3508 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
32sseq1d 3096 . 2 (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4 vex 2663 . . 3 𝑥 ∈ V
54snss 3619 . 2 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
61, 3, 5vtoclbg 2721 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  wss 3041  {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-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-sn 3503
This theorem is referenced by:  snssi  3634  snssd  3635  prssg  3647  ordtri2orexmid  4408  ordtri2or2exmid  4456  relsng  4612  fvimacnvi  5502  fvimacnv  5503  strslfv  11930  isneip  12242  elnei  12248  iscnp4  12314  cnpnei  12315
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