Theorem intmin 3851

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 2733	. . . . 5 |- y e. _V
2	1	elintrab 3843	. . . 4 |- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid 3167	. . . . 5 |- A C_ A
4		sseq2 3171	. . . . . . 7 |- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2 2234	. . . . . . 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 2830	. . . . 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 3153	. 2 |- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub 3849	. . 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 3164	1 |- ( A e. B -> |^| { x e. B | A C_ x } = A )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   C_ wss 3121   |^|cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by:  intmin2  3857  bm2.5ii  4480  onsucmin  4491  cldcls  12908