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.

Step	Hyp	Ref	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 ) )
4	3	biimprd 156	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 11135	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 11273	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3039	. 2 |- ( om C_ A <-> |^| { x | Ind x } C_ A )
8	5, 7	syl6ibr 160	1 |- ( A e. V -> (Ind A -> om C_ A ) )

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