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Theorem bdss 12989
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 𝑥𝐴

Proof of Theorem bdss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bdss.1 . . . 4 BOUNDED 𝐴
21bdeli 12971 . . 3 BOUNDED 𝑦𝐴
32ax-bdal 12943 . 2 BOUNDED𝑦𝑥 𝑦𝐴
4 dfss3 3057 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4bd0r 12950 1 BOUNDED 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1465  wral 2393  wss 3041  BOUNDED wbd 12937  BOUNDED wbdc 12965
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdal 12943
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-in 3047  df-ss 3054  df-bdc 12966
This theorem is referenced by:  bdeq0  12992  bdcpw  12994  bdvsn  12999  bdop  13000  bdeqsuc  13006  bj-nntrans  13076  bj-omtrans  13081
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