Theorem ssunieq 4943

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 4937	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
2		unissb 4939	. . . 4 ⊢ (∪ 𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴)
3	2	biimpri 228	. . 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 3999	. 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 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∪ cuni 4907

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908

This theorem is referenced by:  unimax  4944  shsspwh  31265