Theorem bj-omssind 16070

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 2350	. . 3 |- F/_ x A
2		nfv 1552	. . 3 |- F/ xInd A
3		bj-indeq 16064	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 158	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 15926	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 16068	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3227	. 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 2178   {cab 2193   C_ wss 3174   |^|cint 3899   omcom 4656  Ind wind 16061

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-int 3900  df-iom 4657  df-bj-ind 16062

This theorem is referenced by:  bj-om  16072  peano5set  16075