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Theorem bj-omssind 15124
Description:  om is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-omssind  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )

Proof of Theorem bj-omssind
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nfcv 2332 . . 3  |-  F/_ x A
2 nfv 1539 . . 3  |-  F/ xInd  A
3 bj-indeq 15118 . . . 4  |-  ( x  =  A  ->  (Ind  x 
<-> Ind 
A ) )
43biimprd 158 . . 3  |-  ( x  =  A  ->  (Ind  A  -> Ind  x ) )
51, 2, 4bj-intabssel1 14979 . 2  |-  ( A  e.  V  ->  (Ind  A  ->  |^| { x  | Ind  x }  C_  A
) )
6 bj-dfom 15122 . . 3  |-  om  =  |^| { x  | Ind  x }
76sseq1i 3196 . 2  |-  ( om  C_  A  <->  |^| { x  | Ind  x }  C_  A
)
85, 7imbitrrdi 162 1  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   {cab 2175    C_ wss 3144   |^|cint 3859   omcom 4604  Ind wind 15115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-int 3860  df-iom 4605  df-bj-ind 15116
This theorem is referenced by:  bj-om  15126  peano5set  15129
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