Theorem bj-ssom 10060

Description: A characterization of subclasses of om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ssom	|- ( A. x (Ind x -> A C_ x ) <-> A C_ om )

Distinct variable group:   x, A

Proof of Theorem bj-ssom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3631	. . 3 |- ( A C_ |^| { y | Ind y } <-> A. x e. { y | Ind y } A C_ x )
2		df-ral 2311	. . 3 |- ( A. x e. { y | Ind y } A C_ x <-> A. x ( x e. { y | Ind y } -> A C_ x ) )
3		vex 2560	. . . . . 6 |- x e. _V
4		bj-indeq 10053	. . . . . 6 |- ( y = x -> (Ind y <-> Ind x ) )
5	3, 4	elab 2687	. . . . 5 |- ( x e. { y | Ind y } <-> Ind x )
6	5	imbi1i 227	. . . 4 |- ( ( x e. { y | Ind y } -> A C_ x ) <-> (Ind x -> A C_ x ) )
7	6	albii 1359	. . 3 |- ( A. x ( x e. { y | Ind y } -> A C_ x ) <-> A. x (Ind x -> A C_ x ) )
8	1, 2, 7	3bitrri 196	. 2 |- ( A. x (Ind x -> A C_ x ) <-> A C_ |^| { y | Ind y } )
9		bj-dfom 10057	. . . 4 |- om = |^| { y | Ind y }
10	9	eqcomi 2044	. . 3 |- |^| { y | Ind y } = om
11	10	sseq2i 2970	. 2 |- ( A C_ |^| { y | Ind y } <-> A C_ om )
12	8, 11	bitri 173	1 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   e. wcel 1393   {cab 2026   A.wral 2306   C_ wss 2917   |^|cint 3615   omcom 4313  Ind wind 10050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-iom 4314  df-bj-ind 10051

This theorem is referenced by:  bj-om  10061