Theorem bj-omssind 10730

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 2219	. . 3 |- F/_ x A
2		nfv 1461	. . 3 |- F/ xInd A
3		bj-indeq 10724	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 156	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 10600	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 10728	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3023	. 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 1284   e. wcel 1433   {cab 2067   C_ wss 2973   |^|cint 3636   omcom 4331  Ind wind 10721

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-int 3637  df-iom 4332  df-bj-ind 10722

This theorem is referenced by:  bj-om  10732  peano5set  10735