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 ∈ B → ∩{x ∈ B ∣ A ⊆ 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 ∈ V
2	1	elintrab 3939	. . . 4 ⊢ (y ∈ ∩{x ∈ B ∣ A ⊆ x} ↔ ∀x ∈ B (A ⊆ x → y ∈ x))
3		ssid 3291	. . . . 5 ⊢ A ⊆ A
4		sseq2 3294	. . . . . . 7 ⊢ (x = A → (A ⊆ x ↔ A ⊆ A))
5		eleq2 2414	. . . . . . 7 ⊢ (x = A → (y ∈ x ↔ y ∈ A))
6	4, 5	imbi12d 311	. . . . . 6 ⊢ (x = A → ((A ⊆ x → y ∈ x) ↔ (A ⊆ A → y ∈ A)))
7	6	rspcv 2952	. . . . 5 ⊢ (A ∈ B → (∀x ∈ B (A ⊆ x → y ∈ x) → (A ⊆ A → y ∈ A)))
8	3, 7	mpii 39	. . . 4 ⊢ (A ∈ B → (∀x ∈ B (A ⊆ x → y ∈ x) → y ∈ A))
9	2, 8	syl5bi 208	. . 3 ⊢ (A ∈ B → (y ∈ ∩{x ∈ B ∣ A ⊆ x} → y ∈ A))
10	9	ssrdv 3279	. 2 ⊢ (A ∈ B → ∩{x ∈ B ∣ A ⊆ x} ⊆ A)
11		ssintub 3945	. . 3 ⊢ A ⊆ ∩{x ∈ B ∣ A ⊆ x}
12	11	a1i 10	. 2 ⊢ (A ∈ B → A ⊆ ∩{x ∈ B ∣ A ⊆ x})
13	10, 12	eqssd 3290	1 ⊢ (A ∈ B → ∩{x ∈ B ∣ A ⊆ x} = A)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {crab 2619   ⊆ 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