Theorem unissb 3922

Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
unissb	⊢ (∪A ⊆ B ↔ ∀x ∈ A x ⊆ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem unissb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3895	. . . . . 6 ⊢ (y ∈ ∪A ↔ ∃x(y ∈ x ∧ x ∈ A))
2	1	imbi1i 315	. . . . 5 ⊢ ((y ∈ ∪A → y ∈ B) ↔ (∃x(y ∈ x ∧ x ∈ A) → y ∈ B))
3		19.23v 1891	. . . . 5 ⊢ (∀x((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ (∃x(y ∈ x ∧ x ∈ A) → y ∈ B))
4	2, 3	bitr4i 243	. . . 4 ⊢ ((y ∈ ∪A → y ∈ B) ↔ ∀x((y ∈ x ∧ x ∈ A) → y ∈ B))
5	4	albii 1566	. . 3 ⊢ (∀y(y ∈ ∪A → y ∈ B) ↔ ∀y∀x((y ∈ x ∧ x ∈ A) → y ∈ B))
6		alcom 1737	. . . 4 ⊢ (∀y∀x((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ ∀x∀y((y ∈ x ∧ x ∈ A) → y ∈ B))
7		19.21v 1890	. . . . . 6 ⊢ (∀y(x ∈ A → (y ∈ x → y ∈ B)) ↔ (x ∈ A → ∀y(y ∈ x → y ∈ B)))
8		impexp 433	. . . . . . . 8 ⊢ (((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ (y ∈ x → (x ∈ A → y ∈ B)))
9		bi2.04 350	. . . . . . . 8 ⊢ ((y ∈ x → (x ∈ A → y ∈ B)) ↔ (x ∈ A → (y ∈ x → y ∈ B)))
10	8, 9	bitri 240	. . . . . . 7 ⊢ (((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ (x ∈ A → (y ∈ x → y ∈ B)))
11	10	albii 1566	. . . . . 6 ⊢ (∀y((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ ∀y(x ∈ A → (y ∈ x → y ∈ B)))
12		dfss2 3263	. . . . . . 7 ⊢ (x ⊆ B ↔ ∀y(y ∈ x → y ∈ B))
13	12	imbi2i 303	. . . . . 6 ⊢ ((x ∈ A → x ⊆ B) ↔ (x ∈ A → ∀y(y ∈ x → y ∈ B)))
14	7, 11, 13	3bitr4i 268	. . . . 5 ⊢ (∀y((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ (x ∈ A → x ⊆ B))
15	14	albii 1566	. . . 4 ⊢ (∀x∀y((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ ∀x(x ∈ A → x ⊆ B))
16	6, 15	bitri 240	. . 3 ⊢ (∀y∀x((y ∈ x ∧ x ∈ A) → y ∈ B) ↔ ∀x(x ∈ A → x ⊆ B))
17	5, 16	bitri 240	. 2 ⊢ (∀y(y ∈ ∪A → y ∈ B) ↔ ∀x(x ∈ A → x ⊆ B))
18		dfss2 3263	. 2 ⊢ (∪A ⊆ B ↔ ∀y(y ∈ ∪A → y ∈ B))
19		df-ral 2620	. 2 ⊢ (∀x ∈ A x ⊆ B ↔ ∀x(x ∈ A → x ⊆ B))
20	17, 18, 19	3bitr4i 268	1 ⊢ (∪A ⊆ B ↔ ∀x ∈ A x ⊆ B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258  ∪cuni 3892

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  uniss2  3923  ssunieq  3925  sspwuni  4052  pwssb  4053