Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omssind Unicode version

Theorem bj-omssind 11275
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 2225 . . 3  |-  F/_ x A
2 nfv 1464 . . 3  |-  F/ xInd  A
3 bj-indeq 11269 . . . 4  |-  ( x  =  A  ->  (Ind  x 
<-> Ind 
A ) )
43biimprd 156 . . 3  |-  ( x  =  A  ->  (Ind  A  -> Ind  x ) )
51, 2, 4bj-intabssel1 11135 . 2  |-  ( A  e.  V  ->  (Ind  A  ->  |^| { x  | Ind  x }  C_  A
) )
6 bj-dfom 11273 . . 3  |-  om  =  |^| { x  | Ind  x }
76sseq1i 3039 . 2  |-  ( om  C_  A  <->  |^| { x  | Ind  x }  C_  A
)
85, 7syl6ibr 160 1  |-  ( A  e.  V  ->  (Ind  A  ->  om  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   {cab 2071    C_ wss 2988   |^|cint 3671   omcom 4378  Ind wind 11266
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-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-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
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-v 2617  df-in 2994  df-ss 3001  df-int 3672  df-iom 4379  df-bj-ind 11267
This theorem is referenced by:  bj-om  11277  peano5set  11280
  Copyright terms: Public domain W3C validator