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Theorem bj-omex2 14768
Description: Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 14732 (see bj-2inf 14729 for the equivalence of the latter with bj-omex 14733). (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 14767 . . 3  |-  E. a A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )
2 vex 2742 . . . 4  |-  a  e. 
_V
3 bdcv 14639 . . . . 5  |- BOUNDED  a
43bj-inf2vn 14765 . . . 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 1602 . 2  |-  E. a 
a  =  om
76issetri 2748 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 708   A.wal 1351    = wceq 1353    e. wcel 2148   E.wrex 2456   _Vcvv 2739   (/)c0 3424   suc csuc 4367   omcom 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14604  ax-bdim 14605  ax-bdor 14607  ax-bdex 14610  ax-bdeq 14611  ax-bdel 14612  ax-bdsb 14613  ax-bdsep 14675  ax-bdsetind 14759  ax-inf2 14767
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14632  df-bj-ind 14718
This theorem is referenced by: (None)
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