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

Theorem bdcuni 9265
Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
Assertion
Ref Expression
bdcuni BOUNDED x

Proof of Theorem bdcuni
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdel 9210 . . . . 5 BOUNDED y z
21ax-bdex 9208 . . . 4 BOUNDED z x y z
32bdcab 9238 . . 3 BOUNDED {yz x y z}
4 df-rex 2306 . . . . 5 (z x y zz(z x y z))
5 exancom 1496 . . . . 5 (z(z x y z) ↔ z(y z z x))
64, 5bitri 173 . . . 4 (z x y zz(y z z x))
76abbii 2150 . . 3 {yz x y z} = {yz(y z z x)}
83, 7bdceqi 9232 . 2 BOUNDED {yz(y z z x)}
9 df-uni 3572 . 2 x = {yz(y z z x)}
108, 9bdceqir 9233 1 BOUNDED x
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378  {cab 2023  wrex 2301   cuni 3571  BOUNDED wbdc 9229
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 9202  ax-bdex 9208  ax-bdel 9210  ax-bdsb 9211
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306  df-uni 3572  df-bdc 9230
This theorem is referenced by:  bj-uniex2  9301
  Copyright terms: Public domain W3C validator