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Theorem bdss 13146
Description: The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdss.1  |- BOUNDED  A
Assertion
Ref Expression
bdss  |- BOUNDED  x  C_  A

Proof of Theorem bdss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bdss.1 . . . 4  |- BOUNDED  A
21bdeli 13128 . . 3  |- BOUNDED  y  e.  A
32ax-bdal 13100 . 2  |- BOUNDED  A. y  e.  x  y  e.  A
4 dfss3 3087 . 2  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
53, 4bd0r 13107 1  |- BOUNDED  x  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   A.wral 2416    C_ wss 3071  BOUNDED wbd 13094  BOUNDED wbdc 13122
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13095  ax-bdal 13100
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-in 3077  df-ss 3084  df-bdc 13123
This theorem is referenced by:  bdeq0  13149  bdcpw  13151  bdvsn  13156  bdop  13157  bdeqsuc  13163  bj-nntrans  13233  bj-omtrans  13238
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