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Theorem bj-om 11262
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 11259 . . . 4  |- Ind  om
2 bj-indeq 11254 . . . 4  |-  ( A  =  om  ->  (Ind  A 
<-> Ind 
om ) )
31, 2mpbiri 166 . . 3  |-  ( A  =  om  -> Ind  A )
4 vex 2618 . . . . . 6  |-  x  e. 
_V
5 bj-omssind 11260 . . . . . 6  |-  ( x  e.  _V  ->  (Ind  x  ->  om  C_  x ) )
64, 5ax-mp 7 . . . . 5  |-  (Ind  x  ->  om  C_  x )
7 sseq1 3036 . . . . 5  |-  ( A  =  om  ->  ( A  C_  x  <->  om  C_  x
) )
86, 7syl5ibr 154 . . . 4  |-  ( A  =  om  ->  (Ind  x  ->  A  C_  x
) )
98alrimiv 1799 . . 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 11261 . . . . . . 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 11260 . . . . 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 3029 . . 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 1285    = wceq 1287    e. wcel 1436   _Vcvv 2615    C_ wss 2988   omcom 4377  Ind wind 11251
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3939  ax-pr 4009  ax-un 4233  ax-bd0 11134  ax-bdor 11137  ax-bdex 11140  ax-bdeq 11141  ax-bdel 11142  ax-bdsb 11143  ax-bdsep 11205
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-pr 3438  df-uni 3637  df-int 3672  df-suc 4171  df-iom 4378  df-bdc 11162  df-bj-ind 11252
This theorem is referenced by:  bj-2inf  11263  bj-inf2vn  11299  bj-inf2vn2  11300
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