Theorem unimax 3770

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 3117	. . 3 |- A C_ A
2		sseq1 3120	. . . 4 |- ( x = A -> ( x C_ A <-> A C_ A ) )
3	2	elrab3 2841	. . 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 3120	. . . . 5 |- ( x = y -> ( x C_ A <-> y C_ A ) )
6	5	elrab 2840	. . . 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 2485	. 2 |- A. y e. { x e. B | x C_ A } y C_ A
9		ssunieq 3769	. . 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 2145	. 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 409	1 |- ( A e. B -> U. { x e. B | x C_ A } = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   {crab 2420   C_ wss 3071   U.cuni 3736

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737

This theorem is referenced by:  onuniss2  4428