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Theorem bdss 13481
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 13463 . . 3 BOUNDED 𝑦𝐴
32ax-bdal 13435 . 2 BOUNDED𝑦𝑥 𝑦𝐴
4 dfss3 3118 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
53, 4bd0r 13442 1 BOUNDED 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2128  wral 2435  wss 3102  BOUNDED wbd 13429  BOUNDED wbdc 13457
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bd0 13430  ax-bdal 13435
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-in 3108  df-ss 3115  df-bdc 13458
This theorem is referenced by:  bdeq0  13484  bdcpw  13486  bdvsn  13491  bdop  13492  bdeqsuc  13498  bj-nntrans  13568  bj-omtrans  13573
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