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Theorem snssg 3581
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 2151 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 sneq 3463 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
32sseq1d 3056 . 2 (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4 vex 2625 . . 3 𝑥 ∈ V
54snss 3574 . 2 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
61, 3, 5vtoclbg 2683 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1290  wcel 1439  wss 3002  {csn 3452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-in 3008  df-ss 3015  df-sn 3458
This theorem is referenced by:  snssi  3589  snssd  3590  prssg  3602  ordtri2orexmid  4354  ordtri2or2exmid  4402  relsng  4556  fvimacnvi  5429  fvimacnv  5430  strslfv  11601  isneip  11909  elnei  11915
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