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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
Assertion
Ref Expression
bdss BOUNDED

Proof of Theorem bdss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bdss.1 . . . 4 BOUNDED
21bdeli 13128 . . 3 BOUNDED
32ax-bdal 13100 . 2 BOUNDED
4 dfss3 3087 . 2
53, 4bd0r 13107 1 BOUNDED
 Colors of variables: wff set class Syntax hints:   wcel 1480  wral 2416   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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