Theorem intmin 3828

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 2715	. . . . 5 |- y e. _V
2	1	elintrab 3820	. . . 4 |- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid 3148	. . . . 5 |- A C_ A
4		sseq2 3152	. . . . . . 7 |- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2 2221	. . . . . . 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 2812	. . . . 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 3134	. 2 |- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub 3826	. . 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 3145	1 |- ( A e. B -> |^| { x e. B | A C_ x } = A )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   C_ wss 3102   |^|cint 3808

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-in 3108  df-ss 3115  df-int 3809

This theorem is referenced by:  intmin2  3834  bm2.5ii  4456  onsucmin  4467  cldcls  12556