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Theorem bj-omex2 13552
Description: Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 13516 (see bj-2inf 13513 for the equivalence of the latter with bj-omex 13517). (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 13551 . . 3  |-  E. a A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )
2 vex 2715 . . . 4  |-  a  e. 
_V
3 bdcv 13423 . . . . 5  |- BOUNDED  a
43bj-inf2vn 13549 . . . 4  |-  ( a  e.  _V  ->  ( A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )  -> 
a  =  om )
)
52, 4ax-mp 5 . . 3  |-  ( A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )  -> 
a  =  om )
61, 5eximii 1582 . 2  |-  E. a 
a  =  om
76issetri 2721 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 698   A.wal 1333    = wceq 1335    e. wcel 2128   E.wrex 2436   _Vcvv 2712   (/)c0 3394   suc csuc 4325   omcom 4549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4090  ax-pr 4169  ax-un 4393  ax-bd0 13388  ax-bdim 13389  ax-bdor 13391  ax-bdex 13394  ax-bdeq 13395  ax-bdel 13396  ax-bdsb 13397  ax-bdsep 13459  ax-bdsetind 13543  ax-inf2 13551
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4550  df-bdc 13416  df-bj-ind 13502
This theorem is referenced by: (None)
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