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Theorem snexprc 3965
Description: A singleton whose element is a proper class is a set. The 
-.  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
Assertion
Ref Expression
snexprc  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )

Proof of Theorem snexprc
StepHypRef Expression
1 snprc 3462 . . 3  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
21biimpi 117 . 2  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
3 0ex 3911 . 2  |-  (/)  e.  _V
42, 3syl6eqel 2144 1  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1259    e. wcel 1409   _Vcvv 2574   (/)c0 3251   {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-nul 3252  df-sn 3408
This theorem is referenced by: (None)
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