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Theorem bj-om 10917
Description: A set is equal to  om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-om  |-  ( A  e.  V  ->  ( A  =  om  <->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x )
) ) )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem bj-om
StepHypRef Expression
1 bj-omind 10914 . . . 4  |- Ind  om
2 bj-indeq 10909 . . . 4  |-  ( A  =  om  ->  (Ind  A 
<-> Ind 
om ) )
31, 2mpbiri 166 . . 3  |-  ( A  =  om  -> Ind  A )
4 vex 2605 . . . . . 6  |-  x  e. 
_V
5 bj-omssind 10915 . . . . . 6  |-  ( x  e.  _V  ->  (Ind  x  ->  om  C_  x ) )
64, 5ax-mp 7 . . . . 5  |-  (Ind  x  ->  om  C_  x )
7 sseq1 3021 . . . . 5  |-  ( A  =  om  ->  ( A  C_  x  <->  om  C_  x
) )
86, 7syl5ibr 154 . . . 4  |-  ( A  =  om  ->  (Ind  x  ->  A  C_  x
) )
98alrimiv 1796 . . 3  |-  ( A  =  om  ->  A. x
(Ind  x  ->  A  C_  x ) )
103, 9jca 300 . 2  |-  ( A  =  om  ->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x
) ) )
11 bj-ssom 10916 . . . . . . 7  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
1211biimpi 118 . . . . . 6  |-  ( A. x (Ind  x  ->  A 
C_  x )  ->  A  C_  om )
1312adantl 271 . . . . 5  |-  ( (Ind  A  /\  A. x
(Ind  x  ->  A  C_  x ) )  ->  A  C_  om )
1413a1i 9 . . . 4  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  A  C_  om )
)
15 bj-omssind 10915 . . . . 5  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )
1615adantrd 273 . . . 4  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  om  C_  A ) )
1714, 16jcad 301 . . 3  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  ( A  C_  om 
/\  om  C_  A ) ) )
18 eqss 3015 . . 3  |-  ( A  =  om  <->  ( A  C_ 
om  /\  om  C_  A
) )
1917, 18syl6ibr 160 . 2  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  A  =  om ) )
2010, 19impbid2 141 1  |-  ( A  e.  V  ->  ( A  =  om  <->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   _Vcvv 2602    C_ wss 2974   omcom 4339  Ind wind 10906
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 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10789  ax-bdor 10792  ax-bdex 10795  ax-bdeq 10796  ax-bdel 10797  ax-bdsb 10798  ax-bdsep 10860
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10817  df-bj-ind 10907
This theorem is referenced by:  bj-2inf  10918  bj-inf2vn  10954  bj-inf2vn2  10955
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