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Theorem bdvsn 15112
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 15108 . . . 4  |- BOUNDED  { y }
21bdss 15102 . . 3  |- BOUNDED  x  C_  { y }
3 bdcv 15086 . . . 4  |- BOUNDED  x
43bdsnss 15111 . . 3  |- BOUNDED  { y }  C_  x
52, 4ax-bdan 15053 . 2  |- BOUNDED  ( x  C_  { y }  /\  { y }  C_  x )
6 eqss 3185 . 2  |-  ( x  =  { y }  <-> 
( x  C_  { y }  /\  { y }  C_  x )
)
75, 6bd0r 15063 1  |- BOUNDED  x  =  {
y }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    C_ wss 3144   {csn 3610  BOUNDED wbd 15050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15051  ax-bdan 15053  ax-bdal 15056  ax-bdeq 15058  ax-bdel 15059  ax-bdsb 15060
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3616  df-bdc 15079
This theorem is referenced by:  bdop  15113
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