Theorem bj-omssind 16298

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 2372	. . 3 |- F/_ x A
2		nfv 1574	. . 3 |- F/ xInd A
3		bj-indeq 16292	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 158	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 16154	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 16296	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3250	. 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 1395   e. wcel 2200   {cab 2215   C_ wss 3197   |^|cint 3923   omcom 4682  Ind wind 16289

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-int 3924  df-iom 4683  df-bj-ind 16290

This theorem is referenced by:  bj-om  16300  peano5set  16303