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Theorem bdvsn 14979
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 14975 . . . 4  |- BOUNDED  { y }
21bdss 14969 . . 3  |- BOUNDED  x  C_  { y }
3 bdcv 14953 . . . 4  |- BOUNDED  x
43bdsnss 14978 . . 3  |- BOUNDED  { y }  C_  x
52, 4ax-bdan 14920 . 2  |- BOUNDED  ( x  C_  { y }  /\  { y }  C_  x )
6 eqss 3182 . 2  |-  ( x  =  { y }  <-> 
( x  C_  { y }  /\  { y }  C_  x )
)
75, 6bd0r 14930 1  |- BOUNDED  x  =  {
y }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1363    C_ wss 3141   {csn 3604  BOUNDED wbd 14917
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bd0 14918  ax-bdan 14920  ax-bdal 14923  ax-bdeq 14925  ax-bdel 14926  ax-bdsb 14927
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-sn 3610  df-bdc 14946
This theorem is referenced by:  bdop  14980
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