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Theorem bdvsn 11122
Description: Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Assertion
Ref Expression
bdvsn  |- BOUNDED  x  =  {
y }
Distinct variable group:    x, y

Proof of Theorem bdvsn
StepHypRef Expression
1 bdcsn 11118 . . . 4  |- BOUNDED  { y }
21bdss 11112 . . 3  |- BOUNDED  x  C_  { y }
3 bdcv 11096 . . . 4  |- BOUNDED  x
43bdsnss 11121 . . 3  |- BOUNDED  { y }  C_  x
52, 4ax-bdan 11063 . 2  |- BOUNDED  ( x  C_  { y }  /\  { y }  C_  x )
6 eqss 3027 . 2  |-  ( x  =  { y }  <-> 
( x  C_  { y }  /\  { y }  C_  x )
)
75, 6bd0r 11073 1  |- BOUNDED  x  =  {
y }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1287    C_ wss 2986   {csn 3425  BOUNDED wbd 11060
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-bd0 11061  ax-bdan 11063  ax-bdal 11066  ax-bdeq 11068  ax-bdel 11069  ax-bdsb 11070
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-in 2992  df-ss 2999  df-sn 3431  df-bdc 11089
This theorem is referenced by:  bdop  11123
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