Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsnss Structured version   GIF version

Theorem bdsnss 9308
Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdsnss.1 BOUNDED A
Assertion
Ref Expression
bdsnss BOUNDED {x} ⊆ A
Distinct variable group:   x,A

Proof of Theorem bdsnss
StepHypRef Expression
1 bdsnss.1 . . 3 BOUNDED A
21bdeli 9281 . 2 BOUNDED x A
3 vex 2554 . . 3 x V
43snss 3485 . 2 (x A ↔ {x} ⊆ A)
52, 4bd0 9259 1 BOUNDED {x} ⊆ A
Colors of variables: wff set class
Syntax hints:   wcel 1390  wss 2911  {csn 3367  BOUNDED wbd 9247  BOUNDED wbdc 9275
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 9248
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373  df-bdc 9276
This theorem is referenced by:  bdvsn  9309  bdeqsuc  9316
  Copyright terms: Public domain W3C validator