Theorem bj-ssom 15666

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 3891	. . 3 |- ( A C_ |^| { y | Ind y } <-> A. x e. { y | Ind y } A C_ x )
2		df-ral 2480	. . 3 |- ( A. x e. { y | Ind y } A C_ x <-> A. x ( x e. { y | Ind y } -> A C_ x ) )
3		vex 2766	. . . . . 6 |- x e. _V
4		bj-indeq 15659	. . . . . 6 |- ( y = x -> (Ind y <-> Ind x ) )
5	3, 4	elab 2908	. . . . 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 1484	. . 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 15663	. . . 4 |- om = |^| { y | Ind y }
10	9	eqcomi 2200	. . 3 |- |^| { y | Ind y } = om
11	10	sseq2i 3211	. 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 1362   e. wcel 2167   {cab 2182   A.wral 2475   C_ wss 3157   |^|cint 3875   omcom 4627  Ind wind 15656

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-int 3876  df-iom 4628  df-bj-ind 15657

This theorem is referenced by:  bj-om  15667