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Theorem snssg 3651
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 2200 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 sneq 3533 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
32sseq1d 3121 . 2 (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4 vex 2684 . . 3 𝑥 ∈ V
54snss 3644 . 2 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
61, 3, 5vtoclbg 2742 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  wss 3066  {csn 3522
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-sn 3528
This theorem is referenced by:  snssi  3659  snssd  3660  prssg  3672  ordtri2orexmid  4433  ordtri2or2exmid  4481  relsng  4637  fvimacnvi  5527  fvimacnv  5528  strslfv  11988  isneip  12300  elnei  12306  iscnp4  12372  cnpnei  12373
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