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Theorem snssg 3844
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 2413 . 2 (x = A → (x BA B))
2 sneq 3744 . . 3 (x = A → {x} = {A})
32sseq1d 3298 . 2 (x = A → ({x} B ↔ {A} B))
4 vex 2862 . . 3 x V
54snss 3838 . 2 (x B ↔ {x} B)
61, 3, 5vtoclbg 2915 1 (A V → (A B ↔ {A} B))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   = wceq 1642   wcel 1710   wss 3257  {csn 3737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-sn 3741
This theorem is referenced by:  snssi  3852  snssd  3853  prssg  3862  snelpwg  4114  elssetkg  4269  nnadjoinpw  4521  spacid  6285
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