Theorem bj-omssind 13817

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 2308	. . 3 |- F/_ x A
2		nfv 1516	. . 3 |- F/ xInd A
3		bj-indeq 13811	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 157	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 13671	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 13815	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3168	. 2 |- ( om C_ A <-> |^| { x | Ind x } C_ A )
8	5, 7	syl6ibr 161	1 |- ( A e. V -> (Ind A -> om C_ A ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   {cab 2151   C_ wss 3116   |^|cint 3824   omcom 4567  Ind wind 13808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-int 3825  df-iom 4568  df-bj-ind 13809

This theorem is referenced by:  bj-om  13819  peano5set  13822