Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdss Unicode version

Theorem bdss 11755
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 11737 . . 3  |- BOUNDED  y  e.  A
32ax-bdal 11709 . 2  |- BOUNDED  A. y  e.  x  y  e.  A
4 dfss3 3015 . 2  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
53, 4bd0r 11716 1  |- BOUNDED  x  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1438   A.wral 2359    C_ wss 2999  BOUNDED wbd 11703  BOUNDED wbdc 11731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11704  ax-bdal 11709
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-in 3005  df-ss 3012  df-bdc 11732
This theorem is referenced by:  bdeq0  11758  bdcpw  11760  bdvsn  11765  bdop  11766  bdeqsuc  11772  bj-nntrans  11846  bj-omtrans  11851
  Copyright terms: Public domain W3C validator