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Theorem bdsnss 11107
Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdsnss.1  |- BOUNDED  A
Assertion
Ref Expression
bdsnss  |- BOUNDED  { x }  C_  A
Distinct variable group:    x, A

Proof of Theorem bdsnss
StepHypRef Expression
1 bdsnss.1 . . 3  |- BOUNDED  A
21bdeli 11080 . 2  |- BOUNDED  x  e.  A
3 vex 2615 . . 3  |-  x  e. 
_V
43snss 3540 . 2  |-  ( x  e.  A  <->  { x }  C_  A )
52, 4bd0 11058 1  |- BOUNDED  { x }  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1434    C_ wss 2984   {csn 3422  BOUNDED wbd 11046  BOUNDED wbdc 11074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bd0 11047
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-sn 3428  df-bdc 11075
This theorem is referenced by:  bdvsn  11108  bdeqsuc  11115
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