Theorem bj-ssom 10731

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 3652	. . 3 |- ( A C_ |^| { y | Ind y } <-> A. x e. { y | Ind y } A C_ x )
2		df-ral 2353	. . 3 |- ( A. x e. { y | Ind y } A C_ x <-> A. x ( x e. { y | Ind y } -> A C_ x ) )
3		vex 2604	. . . . . 6 |- x e. _V
4		bj-indeq 10724	. . . . . 6 |- ( y = x -> (Ind y <-> Ind x ) )
5	3, 4	elab 2738	. . . . 5 |- ( x e. { y | Ind y } <-> Ind x )
6	5	imbi1i 236	. . . 4 |- ( ( x e. { y | Ind y } -> A C_ x ) <-> (Ind x -> A C_ x ) )
7	6	albii 1399	. . 3 |- ( A. x ( x e. { y | Ind y } -> A C_ x ) <-> A. x (Ind x -> A C_ x ) )
8	1, 2, 7	3bitrri 205	. 2 |- ( A. x (Ind x -> A C_ x ) <-> A C_ |^| { y | Ind y } )
9		bj-dfom 10728	. . . 4 |- om = |^| { y | Ind y }
10	9	eqcomi 2085	. . 3 |- |^| { y | Ind y } = om
11	10	sseq2i 3024	. 2 |- ( A C_ |^| { y | Ind y } <-> A C_ om )
12	8, 11	bitri 182	1 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   A.wral 2348   C_ wss 2973   |^|cint 3636   omcom 4331  Ind wind 10721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-int 3637  df-iom 4332  df-bj-ind 10722

This theorem is referenced by:  bj-om  10732