Theorem unissbOLD 4907

Description: Obsolete version of unissb 4906 as of 28-Dec-2024. (Contributed by NM, 20-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unissbOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4877	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2	1	imbi1i 349	. . . . 5 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
3		19.23v 1942	. . . . 5 ⊢ (∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
4	2, 3	bitr4i 278	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
5	4	albii 1819	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
6		alcom 2160	. . . 4 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵))
7		19.21v 1939	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
8		impexp 450	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)))
9		bi2.04 387	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
10	8, 9	bitri 275	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
11	10	albii 1819	. . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
12		df-ss 3934	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
13	12	imbi2i 336	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
14	7, 11, 13	3bitr4i 303	. . . . 5 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
15	14	albii 1819	. . . 4 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
16	6, 15	bitri 275	. . 3 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
17	5, 16	bitri 275	. 2 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
18		df-ss 3934	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵))
19		df-ral 3046	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵))
20	17, 18, 19	3bitr4i 303	1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-ss 3934  df-uni 4875

This theorem is referenced by: (None)