Theorem bj-omssind 13133

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 2281	. . 3 |- F/_ x A
2		nfv 1508	. . 3 |- F/ xInd A
3		bj-indeq 13127	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 157	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 12997	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 13131	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3123	. 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 1331   e. wcel 1480   {cab 2125   C_ wss 3071   |^|cint 3771   omcom 4504  Ind wind 13124

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-int 3772  df-iom 4505  df-bj-ind 13125

This theorem is referenced by:  bj-om  13135  peano5set  13138