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Theorem bj-om 13554
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 13551 . . . 4  |- Ind  om
2 bj-indeq 13546 . . . 4  |-  ( A  =  om  ->  (Ind  A 
<-> Ind 
om ) )
31, 2mpbiri 167 . . 3  |-  ( A  =  om  -> Ind  A )
4 vex 2715 . . . . . 6  |-  x  e. 
_V
5 bj-omssind 13552 . . . . . 6  |-  ( x  e.  _V  ->  (Ind  x  ->  om  C_  x ) )
64, 5ax-mp 5 . . . . 5  |-  (Ind  x  ->  om  C_  x )
7 sseq1 3151 . . . . 5  |-  ( A  =  om  ->  ( A  C_  x  <->  om  C_  x
) )
86, 7syl5ibr 155 . . . 4  |-  ( A  =  om  ->  (Ind  x  ->  A  C_  x
) )
98alrimiv 1854 . . 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 13553 . . . . . . 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 13552 . . . . 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 3143 . . 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 1333    = wceq 1335    e. wcel 2128   _Vcvv 2712    C_ wss 3102   omcom 4550  Ind wind 13543
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4091  ax-pr 4170  ax-un 4394  ax-bd0 13430  ax-bdor 13433  ax-bdex 13436  ax-bdeq 13437  ax-bdel 13438  ax-bdsb 13439  ax-bdsep 13501
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3774  df-int 3809  df-suc 4332  df-iom 4551  df-bdc 13458  df-bj-ind 13544
This theorem is referenced by:  bj-2inf  13555  bj-inf2vn  13591  bj-inf2vn2  13592
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