Theorem bj-ssom 15909

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 3901	. . 3 |- ( A C_ |^| { y | Ind y } <-> A. x e. { y | Ind y } A C_ x )
2		df-ral 2489	. . 3 |- ( A. x e. { y | Ind y } A C_ x <-> A. x ( x e. { y | Ind y } -> A C_ x ) )
3		vex 2775	. . . . . 6 |- x e. _V
4		bj-indeq 15902	. . . . . 6 |- ( y = x -> (Ind y <-> Ind x ) )
5	3, 4	elab 2917	. . . . 5 |- ( x e. { y | Ind y } <-> Ind x )
6	5	imbi1i 238	. . . 4 |- ( ( x e. { y | Ind y } -> A C_ x ) <-> (Ind x -> A C_ x ) )
7	6	albii 1493	. . 3 |- ( A. x ( x e. { y | Ind y } -> A C_ x ) <-> A. x (Ind x -> A C_ x ) )
8	1, 2, 7	3bitrri 207	. 2 |- ( A. x (Ind x -> A C_ x ) <-> A C_ |^| { y | Ind y } )
9		bj-dfom 15906	. . . 4 |- om = |^| { y | Ind y }
10	9	eqcomi 2209	. . 3 |- |^| { y | Ind y } = om
11	10	sseq2i 3220	. 2 |- ( A C_ |^| { y | Ind y } <-> A C_ om )
12	8, 11	bitri 184	1 |- ( A. x (Ind x -> A C_ x ) <-> A C_ om )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   e. wcel 2176   {cab 2191   A.wral 2484   C_ wss 3166   |^|cint 3885   omcom 4639  Ind wind 15899

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-int 3886  df-iom 4640  df-bj-ind 15900

This theorem is referenced by:  bj-om  15910