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Theorem bdss 9319
 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 9301 . . 3 BOUNDED y A
32ax-bdal 9273 . 2 BOUNDED y x y A
4 dfss3 2929 . 2 (xAy x y A)
53, 4bd0r 9280 1 BOUNDED xA
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  BOUNDED wbd 9267  BOUNDED wbdc 9295 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9268  ax-bdal 9273 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-bdc 9296 This theorem is referenced by:  bdeq0  9322  bdcpw  9324  bdvsn  9329  bdop  9330  bdeqsuc  9336  bj-nntrans  9411  bj-omtrans  9416
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