Theorem unimax 4473

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 3624	. . 3 |- A C_ A
2		sseq1 3626	. . . 4 |- ( x = A -> ( x C_ A <-> A C_ A ) )
3	2	elrab3 3364	. . 3 |- ( A e. B -> ( A e. { x e. B | x C_ A } <-> A C_ A ) )
4	1, 3	mpbiri 248	. 2 |- ( A e. B -> A e. { x e. B | x C_ A } )
5		sseq1 3626	. . . . 5 |- ( x = y -> ( x C_ A <-> y C_ A ) )
6	5	elrab 3363	. . . 4 |- ( y e. { x e. B | x C_ A } <-> ( y e. B /\ y C_ A ) )
7	6	simprbi 480	. . 3 |- ( y e. { x e. B | x C_ A } -> y C_ A )
8	7	rgen 2922	. 2 |- A. y e. { x e. B | x C_ A } y C_ A
9		ssunieq 4472	. . 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 2628	. 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 694	1 |- ( A e. B -> U. { x e. B | x C_ A } = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by:  lssuni  18940  chsupid  28271  shatomistici  29220  lssats  34299  lpssat  34300  lssatle  34302  lssat  34303