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Theorem bj-om 13972
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 13969 . . . 4  |- Ind  om
2 bj-indeq 13964 . . . 4  |-  ( A  =  om  ->  (Ind  A 
<-> Ind 
om ) )
31, 2mpbiri 167 . . 3  |-  ( A  =  om  -> Ind  A )
4 vex 2733 . . . . . 6  |-  x  e. 
_V
5 bj-omssind 13970 . . . . . 6  |-  ( x  e.  _V  ->  (Ind  x  ->  om  C_  x ) )
64, 5ax-mp 5 . . . . 5  |-  (Ind  x  ->  om  C_  x )
7 sseq1 3170 . . . . 5  |-  ( A  =  om  ->  ( A  C_  x  <->  om  C_  x
) )
86, 7syl5ibr 155 . . . 4  |-  ( A  =  om  ->  (Ind  x  ->  A  C_  x
) )
98alrimiv 1867 . . 3  |-  ( A  =  om  ->  A. x
(Ind  x  ->  A  C_  x ) )
103, 9jca 304 . 2  |-  ( A  =  om  ->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x
) ) )
11 bj-ssom 13971 . . . . . . 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 275 . . . . 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 13970 . . . . 5  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )
1615adantrd 277 . . . 4  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  om  C_  A ) )
1714, 16jcad 305 . . 3  |-  ( A  e.  V  ->  (
(Ind  A  /\  A. x (Ind  x  ->  A 
C_  x ) )  ->  ( A  C_  om 
/\  om  C_  A ) ) )
18 eqss 3162 . . 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 1346    = wceq 1348    e. wcel 2141   _Vcvv 2730    C_ wss 3121   omcom 4574  Ind wind 13961
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-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919
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:  bj-2inf  13973  bj-inf2vn  14009  bj-inf2vn2  14010
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