Theorem intmin 3791

Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
intmin	|- ( A e. B -> |^| { x e. B | A C_ x } = A )

Distinct variable groups:   x, A   x, B

Proof of Theorem intmin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2689	. . . . 5 |- y e. _V
2	1	elintrab 3783	. . . 4 |- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid 3117	. . . . 5 |- A C_ A
4		sseq2 3121	. . . . . . 7 |- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2 2203	. . . . . . 7 |- ( x = A -> ( y e. x <-> y e. A ) )
6	4, 5	imbi12d 233	. . . . . 6 |- ( x = A -> ( ( A C_ x -> y e. x ) <-> ( A C_ A -> y e. A ) ) )
7	6	rspcv 2785	. . . . 5 |- ( A e. B -> ( A. x e. B ( A C_ x -> y e. x ) -> ( A C_ A -> y e. A ) ) )
8	3, 7	mpii 44	. . . 4 |- ( A e. B -> ( A. x e. B ( A C_ x -> y e. x ) -> y e. A ) )
9	2, 8	syl5bi 151	. . 3 |- ( A e. B -> ( y e. |^| { x e. B | A C_ x } -> y e. A ) )
10	9	ssrdv 3103	. 2 |- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub 3789	. . 3 |- A C_ |^| { x e. B | A C_ x }
12	11	a1i 9	. 2 |- ( A e. B -> A C_ |^| { x e. B | A C_ x } )
13	10, 12	eqssd 3114	1 |- ( A e. B -> |^| { x e. B | A C_ x } = A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   A.wral 2416   {crab 2420   C_ wss 3071   |^|cint 3771

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-int 3772

This theorem is referenced by:  intmin2  3797  bm2.5ii  4412  onsucmin  4423  cldcls  12283