Theorem intmin 3799

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 2692	. . . . 5 |- y e. _V
2	1	elintrab 3791	. . . 4 |- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid 3122	. . . . 5 |- A C_ A
4		sseq2 3126	. . . . . . 7 |- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2 2204	. . . . . . 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 2789	. . . . 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 3108	. 2 |- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub 3797	. . 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 3119	1 |- ( A e. B -> |^| { x e. B | A C_ x } = A )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   A.wral 2417   {crab 2421   C_ wss 3076   |^|cint 3779

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089  df-int 3780

This theorem is referenced by:  intmin2  3805  bm2.5ii  4420  onsucmin  4431  cldcls  12322