Theorem ssunieq 4876

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 4871	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
2		unissb 4873	. . . 4 ⊢ (∪ 𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴)
3	2	biimpri 227	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴)
4	1, 3	anim12i 613	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴))
5		eqss 3936	. 2 ⊢ (𝐴 = ∪ 𝐵 ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴))
6	4, 5	sylibr 233	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840

This theorem is referenced by:  unimax  4877  shsspwh  29608