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Theorem snssg 3491
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 (A 𝑉 → (A B ↔ {A} ⊆ B))

Proof of Theorem snssg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . 2 (x = A → (x BA B))
2 sneq 3378 . . 3 (x = A → {x} = {A})
32sseq1d 2966 . 2 (x = A → ({x} ⊆ B ↔ {A} ⊆ B))
4 vex 2554 . . 3 x V
54snss 3485 . 2 (x B ↔ {x} ⊆ B)
61, 3, 5vtoclbg 2608 1 (A 𝑉 → (A B ↔ {A} ⊆ B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wss 2911  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373
This theorem is referenced by:  snssi  3499  snssd  3500  prssg  3512  ordtri2orexmid  4211  fvimacnvi  5224  fvimacnv  5225
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