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Theorem bdeq0 13170
Description: Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
bdeq0  |- BOUNDED  x  =  (/)

Proof of Theorem bdeq0
StepHypRef Expression
1 bdcnul 13168 . . 3  |- BOUNDED  (/)
21bdss 13167 . 2  |- BOUNDED  x  C_  (/)
3 0ss 3401 . . 3  |-  (/)  C_  x
4 eqss 3112 . . 3  |-  ( x  =  (/)  <->  ( x  C_  (/) 
/\  (/)  C_  x )
)
53, 4mpbiran2 925 . 2  |-  ( x  =  (/)  <->  x  C_  (/) )
62, 5bd0r 13128 1  |- BOUNDED  x  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    C_ wss 3071   (/)c0 3363  BOUNDED wbd 13115
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-in1 603  ax-in2 604  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  ax-bdim 13117  ax-bdn 13120  ax-bdal 13121  ax-bdeq 13123
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-bdc 13144
This theorem is referenced by:  bj-bd0el  13171  bj-nn0suc0  13253
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