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Theorem bdss 11097
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 11079 . . 3  |- BOUNDED  y  e.  A
32ax-bdal 11051 . 2  |- BOUNDED  A. y  e.  x  y  e.  A
4 dfss3 3000 . 2  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
53, 4bd0r 11058 1  |- BOUNDED  x  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   A.wral 2353    C_ wss 2984  BOUNDED wbd 11045  BOUNDED wbdc 11073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bd0 11046  ax-bdal 11051
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-in 2990  df-ss 2997  df-bdc 11074
This theorem is referenced by:  bdeq0  11100  bdcpw  11102  bdvsn  11107  bdop  11108  bdeqsuc  11114  bj-nntrans  11188  bj-omtrans  11193
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