Theorem intmin 3948

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 2805	. . . . 5 |- y e. _V
2	1	elintrab 3940	. . . 4 |- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid 3247	. . . . 5 |- A C_ A
4		sseq2 3251	. . . . . . 7 |- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2 2295	. . . . . . 7 |- ( x = A -> ( y e. x <-> y e. A ) )
6	4, 5	imbi12d 234	. . . . . 6 |- ( x = A -> ( ( A C_ x -> y e. x ) <-> ( A C_ A -> y e. A ) ) )
7	6	rspcv 2906	. . . . 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	biimtrid 152	. . 3 |- ( A e. B -> ( y e. |^| { x e. B | A C_ x } -> y e. A ) )
10	9	ssrdv 3233	. 2 |- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub 3946	. . 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 3244	1 |- ( A e. B -> |^| { x e. B | A C_ x } = A )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   C_ wss 3200   |^|cint 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213  df-int 3929

This theorem is referenced by:  intmin2  3954  bm2.5ii  4594  onsucmin  4605  lspid  14410  cldcls  14837