Theorem unimax 3823

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 3162	. . 3 |- A C_ A
2		sseq1 3165	. . . 4 |- ( x = A -> ( x C_ A <-> A C_ A ) )
3	2	elrab3 2883	. . 3 |- ( A e. B -> ( A e. { x e. B | x C_ A } <-> A C_ A ) )
4	1, 3	mpbiri 167	. 2 |- ( A e. B -> A e. { x e. B | x C_ A } )
5		sseq1 3165	. . . . 5 |- ( x = y -> ( x C_ A <-> y C_ A ) )
6	5	elrab 2882	. . . 4 |- ( y e. { x e. B | x C_ A } <-> ( y e. B /\ y C_ A ) )
7	6	simprbi 273	. . 3 |- ( y e. { x e. B | x C_ A } -> y C_ A )
8	7	rgen 2519	. 2 |- A. y e. { x e. B | x C_ A } y C_ A
9		ssunieq 3822	. . 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 2171	. 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 410	1 |- ( A e. B -> U. { x e. B | x C_ A } = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   {crab 2448   C_ wss 3116   U.cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790

This theorem is referenced by:  onuniss2  4489