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Theorem bdeqsuc 13006
Description: Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
bdeqsuc  |- BOUNDED  x  =  suc  y
Distinct variable group:    x, y

Proof of Theorem bdeqsuc
StepHypRef Expression
1 bdcsuc 13005 . . . 4  |- BOUNDED  suc  y
21bdss 12989 . . 3  |- BOUNDED  x  C_  suc  y
3 bdcv 12973 . . . . . . 7  |- BOUNDED  x
43bdss 12989 . . . . . 6  |- BOUNDED  y  C_  x
53bdsnss 12998 . . . . . 6  |- BOUNDED  { y }  C_  x
64, 5ax-bdan 12940 . . . . 5  |- BOUNDED  ( y  C_  x  /\  { y }  C_  x )
7 unss 3220 . . . . 5  |-  ( ( y  C_  x  /\  { y }  C_  x
)  <->  ( y  u. 
{ y } ) 
C_  x )
86, 7bd0 12949 . . . 4  |- BOUNDED  ( y  u.  {
y } )  C_  x
9 df-suc 4263 . . . . 5  |-  suc  y  =  ( y  u. 
{ y } )
109sseq1i 3093 . . . 4  |-  ( suc  y  C_  x  <->  ( y  u.  { y } ) 
C_  x )
118, 10bd0r 12950 . . 3  |- BOUNDED  suc  y  C_  x
122, 11ax-bdan 12940 . 2  |- BOUNDED  ( x  C_  suc  y  /\  suc  y  C_  x )
13 eqss 3082 . 2  |-  ( x  =  suc  y  <->  ( x  C_ 
suc  y  /\  suc  y  C_  x ) )
1412, 13bd0r 12950 1  |- BOUNDED  x  =  suc  y
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    u. cun 3039    C_ wss 3041   {csn 3497   suc csuc 4257  BOUNDED wbd 12937
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdan 12940  ax-bdor 12941  ax-bdal 12943  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-suc 4263  df-bdc 12966
This theorem is referenced by:  bj-bdsucel  13007  bj-nn0suc0  13075
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