Theorem bj-omssind 14547

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 2319	. . 3 |- F/_ x A
2		nfv 1528	. . 3 |- F/ xInd A
3		bj-indeq 14541	. . . 4 |- ( x = A -> (Ind x <-> Ind A ) )
4	3	biimprd 158	. . 3 |- ( x = A -> (Ind A -> Ind x ) )
5	1, 2, 4	bj-intabssel1 14402	. 2 |- ( A e. V -> (Ind A -> |^| { x | Ind x } C_ A ) )
6		bj-dfom 14545	. . 3 |- om = |^| { x | Ind x }
7	6	sseq1i 3181	. 2 |- ( om C_ A <-> |^| { x | Ind x } C_ A )
8	5, 7	syl6ibr 162	1 |- ( A e. V -> (Ind A -> om C_ A ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   C_ wss 3129   |^|cint 3844   omcom 4588  Ind wind 14538

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-int 3845  df-iom 4589  df-bj-ind 14539

This theorem is referenced by:  bj-om  14549  peano5set  14552