Theorem bj-omssind 15908

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 2348	. . 3 |- F/_ x A
2		nfv 1551	. . 3 |- F/ xInd A
3		bj-indeq 15902	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 158	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 15763	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 15906	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3219	. 2 |- ( om C_ A <-> |^| { x | Ind x } C_ A )
8	5, 7	imbitrrdi 162	1 |- ( A e. V -> (Ind A -> om C_ A ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {cab 2191   C_ wss 3166   |^|cint 3885   omcom 4639  Ind wind 15899

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-int 3886  df-iom 4640  df-bj-ind 15900

This theorem is referenced by:  bj-om  15910  peano5set  15913