Theorem intmin 3947

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 2863	. . . . 5 |- y e. _V
2	1	elintrab 3939	. . . 4 |- (y e. |^|{x e. B | A C_ x} <-> A.x e. B (A C_ x -> y e. x))
3		ssid 3291	. . . . 5 |- A C_ A
4		sseq2 3294	. . . . . . 7 |- (x = A -> (A C_ x <-> A C_ A))
5		eleq2 2414	. . . . . . 7 |- (x = A -> (y e. x <-> y e. A))
6	4, 5	imbi12d 311	. . . . . 6 |- (x = A -> ((A C_ x -> y e. x) <-> (A C_ A -> y e. A)))
7	6	rspcv 2952	. . . . 5 |- (A e. B -> (A.x e. B (A C_ x -> y e. x) -> (A C_ A -> y e. A)))
8	3, 7	mpii 39	. . . 4 |- (A e. B -> (A.x e. B (A C_ x -> y e. x) -> y e. A))
9	2, 8	syl5bi 208	. . 3 |- (A e. B -> (y e. |^|{x e. B | A C_ x} -> y e. A))
10	9	ssrdv 3279	. 2 |- (A e. B -> |^|{x e. B | A C_ x} C_ A)
11		ssintub 3945	. . 3 |- A C_ |^|{x e. B | A C_ x}
12	11	a1i 10	. 2 |- (A e. B -> A C_ |^|{x e. B | A C_ x})
13	10, 12	eqssd 3290	1 |- (A e. B -> |^|{x e. B | A C_ x} = A)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  A.wral 2615  {crab 2619   C_ wss 3258  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by:  intmin2  3954