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Theorem bdss 14498
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 14480 . . 3  |- BOUNDED  y  e.  A
32ax-bdal 14452 . 2  |- BOUNDED  A. y  e.  x  y  e.  A
4 dfss3 3145 . 2  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
53, 4bd0r 14459 1  |- BOUNDED  x  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 2148   A.wral 2455    C_ wss 3129  BOUNDED wbd 14446  BOUNDED wbdc 14474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14447  ax-bdal 14452
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3135  df-ss 3142  df-bdc 14475
This theorem is referenced by:  bdeq0  14501  bdcpw  14503  bdvsn  14508  bdop  14509  bdeqsuc  14515  bj-nntrans  14585  bj-omtrans  14590
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