Theorem bj-ssom 13971

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 3847	. . 3 |- ( A C_ |^| { y | Ind y } <-> A. x e. { y | Ind y } A C_ x )
2		df-ral 2453	. . 3 |- ( A. x e. { y | Ind y } A C_ x <-> A. x ( x e. { y | Ind y } -> A C_ x ) )
3		vex 2733	. . . . . 6 |- x e. _V
4		bj-indeq 13964	. . . . . 6 |- ( y = x -> (Ind y <-> Ind x ) )
5	3, 4	elab 2874	. . . . 5 |- ( x e. { y | Ind y } <-> Ind x )
6	5	imbi1i 237	. . . 4 |- ( ( x e. { y | Ind y } -> A C_ x ) <-> (Ind x -> A C_ x ) )
7	6	albii 1463	. . 3 |- ( A. x ( x e. { y | Ind y } -> A C_ x ) <-> A. x (Ind x -> A C_ x ) )
8	1, 2, 7	3bitrri 206	. 2 |- ( A. x (Ind x -> A C_ x ) <-> A C_ |^| { y | Ind y } )
9		bj-dfom 13968	. . . 4 |- om = |^| { y | Ind y }
10	9	eqcomi 2174	. . 3 |- |^| { y | Ind y } = om
11	10	sseq2i 3174	. 2 |- ( A C_ |^| { y | Ind y } <-> A C_ om )
12	8, 11	bitri 183	1 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   e. wcel 2141   {cab 2156   A.wral 2448   C_ wss 3121   |^|cint 3831   omcom 4574  Ind wind 13961

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-iom 4575  df-bj-ind 13962

This theorem is referenced by:  bj-om  13972