Theorem unimax 3858

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 3190	. . 3 |- A C_ A
2		sseq1 3193	. . . 4 |- ( x = A -> ( x C_ A <-> A C_ A ) )
3	2	elrab3 2909	. . 3 |- ( A e. B -> ( A e. { x e. B | x C_ A } <-> A C_ A ) )
4	1, 3	mpbiri 168	. 2 |- ( A e. B -> A e. { x e. B | x C_ A } )
5		sseq1 3193	. . . . 5 |- ( x = y -> ( x C_ A <-> y C_ A ) )
6	5	elrab 2908	. . . 4 |- ( y e. { x e. B | x C_ A } <-> ( y e. B /\ y C_ A ) )
7	6	simprbi 275	. . 3 |- ( y e. { x e. B | x C_ A } -> y C_ A )
8	7	rgen 2543	. 2 |- A. y e. { x e. B | x C_ A } y C_ A
9		ssunieq 3857	. . 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 2195	. 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 413	1 |- ( A e. B -> U. { x e. B | x C_ A } = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   {crab 2472   C_ wss 3144   U.cuni 3824

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825

This theorem is referenced by:  onuniss2  4526  lssuni  13640