Theorem unimax 3635

Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)

Assertion

Ref	Expression
unimax	|- ( A e. B -> U. { x e. B | x C_ A } = A )

Distinct variable groups:   x, A   x, B

Proof of Theorem unimax

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3018	. . 3 |- A C_ A
2		sseq1 3020	. . . 4 |- ( x = A -> ( x C_ A <-> A C_ A ) )
3	2	elrab3 2750	. . 3 |- ( A e. B -> ( A e. { x e. B | x C_ A } <-> A C_ A ) )
4	1, 3	mpbiri 166	. 2 |- ( A e. B -> A e. { x e. B | x C_ A } )
5		sseq1 3020	. . . . 5 |- ( x = y -> ( x C_ A <-> y C_ A ) )
6	5	elrab 2749	. . . 4 |- ( y e. { x e. B | x C_ A } <-> ( y e. B /\ y C_ A ) )
7	6	simprbi 269	. . 3 |- ( y e. { x e. B | x C_ A } -> y C_ A )
8	7	rgen 2416	. 2 |- A. y e. { x e. B | x C_ A } y C_ A
9		ssunieq 3634	. . 3 |- ( ( A e. { x e. B | x C_ A } /\ A. y e. { x e. B | x C_ A } y C_ A ) -> A = U. { x e. B | x C_ A } )
10	9	eqcomd 2086	. 2 |- ( ( A e. { x e. B | x C_ A } /\ A. y e. { x e. B | x C_ A } y C_ A ) -> U. { x e. B | x C_ A } = A )
11	4, 8, 10	sylancl 404	1 |- ( A e. B -> U. { x e. B | x C_ A } = A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   {crab 2352   C_ wss 2973   U.cuni 3601

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-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602

This theorem is referenced by:  onuniss2  4256