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Theorem bj-omex2 14012
Description: Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 13976 (see bj-2inf 13973 for the equivalence of the latter with bj-omex 13977). (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 14011 . . 3  |-  E. a A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )
2 vex 2733 . . . 4  |-  a  e. 
_V
3 bdcv 13883 . . . . 5  |- BOUNDED  a
43bj-inf2vn 14009 . . . 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 1595 . 2  |-  E. a 
a  =  om
76issetri 2739 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 703   A.wal 1346    = wceq 1348    e. wcel 2141   E.wrex 2449   _Vcvv 2730   (/)c0 3414   suc csuc 4350   omcom 4574
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdim 13849  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-bdsetind 14003  ax-inf2 14011
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962
This theorem is referenced by: (None)
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