Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omex2 Unicode version

Theorem bj-omex2 11057
Description: Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 11021 (see bj-2inf 11018 for the equivalence of the latter with bj-omex 11022). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-omex2  |-  om  e.  _V

Proof of Theorem bj-omex2
Dummy variables  x  y  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 11056 . . 3  |-  E. a A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )
2 vex 2613 . . . 4  |-  a  e. 
_V
3 bdcv 10924 . . . . 5  |- BOUNDED  a
43bj-inf2vn 11054 . . . 4  |-  ( a  e.  _V  ->  ( A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )  -> 
a  =  om )
)
52, 4ax-mp 7 . . 3  |-  ( A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )  -> 
a  =  om )
61, 5eximii 1534 . 2  |-  E. a 
a  =  om
76issetri 2617 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   E.wrex 2354   _Vcvv 2610   (/)c0 3267   suc csuc 4148   omcom 4359
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3924  ax-pr 3992  ax-un 4216  ax-bd0 10889  ax-bdim 10890  ax-bdor 10892  ax-bdex 10895  ax-bdeq 10896  ax-bdel 10897  ax-bdsb 10898  ax-bdsep 10960  ax-bdsetind 11048  ax-inf2 11056
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-suc 4154  df-iom 4360  df-bdc 10917  df-bj-ind 11007
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator