Theorem unissel 4905

Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
unissel	⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵)

Proof of Theorem unissel

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 ⊆ 𝐵)
2		elssuni 4904	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴)
3	2	adantl 481	. 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴)
4	1, 3	eqssd 3967	1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ 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-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-v 3452  df-ss 3934  df-uni 4875

This theorem is referenced by:  elpwuni  5072  mretopd  22986  toponmre  22987  neiptopuni  23024  filunibas  23775  unidmvol  25449  unicls  33900  carsguni  34306  onintunirab  43223