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Theorem bj-om 12946
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 12943 . . . 4  |- Ind  om
2 bj-indeq 12938 . . . 4  |-  ( A  =  om  ->  (Ind  A 
<-> Ind 
om ) )
31, 2mpbiri 167 . . 3  |-  ( A  =  om  -> Ind  A )
4 vex 2661 . . . . . 6  |-  x  e. 
_V
5 bj-omssind 12944 . . . . . 6  |-  ( x  e.  _V  ->  (Ind  x  ->  om  C_  x ) )
64, 5ax-mp 5 . . . . 5  |-  (Ind  x  ->  om  C_  x )
7 sseq1 3088 . . . . 5  |-  ( A  =  om  ->  ( A  C_  x  <->  om  C_  x
) )
86, 7syl5ibr 155 . . . 4  |-  ( A  =  om  ->  (Ind  x  ->  A  C_  x
) )
98alrimiv 1828 . . 3  |-  ( A  =  om  ->  A. x
(Ind  x  ->  A  C_  x ) )
103, 9jca 302 . 2  |-  ( A  =  om  ->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x
) ) )
11 bj-ssom 12945 . . . . . . 7  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
1211biimpi 119 . . . . . 6  |-  ( A. x (Ind  x  ->  A 
C_  x )  ->  A  C_  om )
1312adantl 273 . . . . 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 12944 . . . . 5  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )
1615adantrd 275 . . . 4  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  om  C_  A ) )
1714, 16jcad 303 . . 3  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  ( A  C_  om 
/\  om  C_  A ) ) )
18 eqss 3080 . . 3  |-  ( A  =  om  <->  ( A  C_ 
om  /\  om  C_  A
) )
1917, 18syl6ibr 161 . 2  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  A  =  om ) )
2010, 19impbid2 142 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 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   _Vcvv 2658    C_ wss 3039   omcom 4472  Ind wind 12935
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12822  ax-bdor 12825  ax-bdex 12828  ax-bdeq 12829  ax-bdel 12830  ax-bdsb 12831  ax-bdsep 12893
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12850  df-bj-ind 12936
This theorem is referenced by:  bj-2inf  12947  bj-inf2vn  12983  bj-inf2vn2  12984
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