Theorem unimax 3926

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 3291	. . 3 |- A C_ A
2		sseq1 3293	. . . 4 |- (x = A -> (x C_ A <-> A C_ A))
3	2	elrab3 2996	. . 3 |- (A e. B -> (A e. {x e. B | x C_ A} <-> A C_ A))
4	1, 3	mpbiri 224	. 2 |- (A e. B -> A e. {x e. B | x C_ A})
5		sseq1 3293	. . . . 5 |- (x = y -> (x C_ A <-> y C_ A))
6	5	elrab 2995	. . . 4 |- (y e. {x e. B | x C_ A} <-> (y e. B /\ y C_ A))
7	6	simprbi 450	. . 3 |- (y e. {x e. B | x C_ A} -> y C_ A)
8	7	rgen 2680	. 2 |- A.y e. {x e. B | x C_ A}y C_ A
9		ssunieq 3925	. . 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 2358	. 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 643	1 |- (A e. B -> U.{x e. B | x C_ A} = A)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  {crab 2619   C_ wss 3258  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by: (None)