Theorem ssunieq 4967

Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Assertion

Ref	Expression
ssunieq	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 4961	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
2		unissb 4963	. . . 4 ⊢ (∪ 𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴)
3	2	biimpri 228	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴)
4	1, 3	anim12i 612	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴))
5		eqss 4024	. 2 ⊢ (𝐴 = ∪ 𝐵 ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴))
6	4, 5	sylibr 234	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-ss 3993  df-uni 4932

This theorem is referenced by:  unimax  4968  shsspwh  31278