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Theorem bdsnss 13176
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 13149 . 2  |- BOUNDED  x  e.  A
3 vex 2689 . . 3  |-  x  e. 
_V
43snss 3649 . 2  |-  ( x  e.  A  <->  { x }  C_  A )
52, 4bd0 13127 1  |- BOUNDED  { x }  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1480    C_ wss 3071   {csn 3527  BOUNDED wbd 13115  BOUNDED wbdc 13143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13116
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-bdc 13144
This theorem is referenced by:  bdvsn  13177  bdeqsuc  13184
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