Theorem intmin 1982

Description: Any member of a class is the smallest of those members that include it.

Assertion

Ref	Expression
intmin	⊢ (A ∈ B → A = ∩{x ∈ B∣A ⊆ x})

Distinct variable group(s):   x,A   x,B

Proof of Theorem intmin

Step	Hyp	Ref	Expression
1		ssid 1519	. . . . . . 7 ⊢ A ⊆ A
2		sseq2 1522	. . . . . . . . 9 ⊢ (x = A → (A ⊆ x ↔ A ⊆ A))
3		eleq2 1150	. . . . . . . . 9 ⊢ (x = A → (y ∈ x ↔ y ∈ A))
4	2, 3	imbi12d 474	. . . . . . . 8 ⊢ (x = A → ((A ⊆ x → y ∈ x) ↔ (A ⊆ A → y ∈ A)))
5	4	rcla4v 1402	. . . . . . 7 ⊢ (∀x ∈ B (A ⊆ x → y ∈ x) → (A ∈ B → (A ⊆ A → y ∈ A)))
6	1, 5	mpii 45	. . . . . 6 ⊢ (∀x ∈ B (A ⊆ x → y ∈ x) → (A ∈ B → y ∈ A))
7	6	com12 13	. . . . 5 ⊢ (A ∈ B → (∀x ∈ B (A ⊆ x → y ∈ x) → y ∈ A))
8		visset 1350	. . . . . 6 ⊢ y ∈ V
9	8	elintrab 1977	. . . . 5 ⊢ (y ∈ ∩{x ∈ B∣A ⊆ x} ↔ ∀x ∈ B (A ⊆ x → y ∈ x))
10	7, 9	syl5ib 181	. . . 4 ⊢ (A ∈ B → (y ∈ ∩{x ∈ B∣A ⊆ x} → y ∈ A))
11	10	ssrdv 1509	. . 3 ⊢ (A ∈ B → ∩{x ∈ B∣A ⊆ x} ⊆ A)
12		ssintub 1981	. . 3 ⊢ A ⊆ ∩{x ∈ B∣A ⊆ x}
13	11, 12	jctil 240	. 2 ⊢ (A ∈ B → (A ⊆ ∩{x ∈ B∣A ⊆ x} ∧ ∩{x ∈ B∣A ⊆ x} ⊆ A))
14		eqss 1516	. 2 ⊢ (A = ∩{x ∈ B∣A ⊆ x} ↔ (A ⊆ ∩{x ∈ B∣A ⊆ x} ∧ ∩{x ∈ B∣A ⊆ x} ⊆ A))
15	13, 14	sylibr 175	1 ⊢ (A ∈ B → A = ∩{x ∈ B∣A ⊆ x})

Syntax hints:   → wi 2   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487  ∩cint 1965

This theorem is referenced by:  intmin2 1984  bm2.5ii 2274  onsucmin 2323  rankonid 3538  rankr1id 3539  ranklon 3540  chsupid 5312  spanid 5318

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208  df-v 1349  df-in 1491  df-ss 1492  df-int 1966

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