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Theorem snssg 3470
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 2078 . 2 (x = A → (x BA B))
2 sneq 3357 . . 3 (x = A → {x} = {A})
32sseq1d 2945 . 2 (x = A → ({x} ⊆ B ↔ {A} ⊆ B))
4 vex 2534 . . 3 x V
54snss 3464 . 2 (x B ↔ {x} ⊆ B)
61, 3, 5vtoclbg 2587 1 (A 𝑉 → (A B ↔ {A} ⊆ B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226   wcel 1370  wss 2890  {csn 3346
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904  df-sn 3352
This theorem is referenced by:  snssi  3478  snssd  3479  prssg  3491  ordtri2orexmid  4191  fvimacnvi  5202  fvimacnv  5203
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