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Theorem bdeq0 12031
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 12029 . . 3  |- BOUNDED  (/)
21bdss 12028 . 2  |- BOUNDED  x  C_  (/)
3 0ss 3325 . . 3  |-  (/)  C_  x
4 eqss 3041 . . 3  |-  ( x  =  (/)  <->  ( x  C_  (/) 
/\  (/)  C_  x )
)
53, 4mpbiran2 888 . 2  |-  ( x  =  (/)  <->  x  C_  (/) )
62, 5bd0r 11989 1  |- BOUNDED  x  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1290    C_ wss 3000   (/)c0 3287  BOUNDED wbd 11976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-bd0 11977  ax-bdim 11978  ax-bdn 11981  ax-bdal 11982  ax-bdeq 11984
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288  df-bdc 12005
This theorem is referenced by:  bj-bd0el  12032  bj-nn0suc0  12118
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