Theorem unissel 3921

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 443	. 2 ⊢ ((∪A ⊆ B ∧ B ∈ A) → ∪A ⊆ B)
2		elssuni 3920	. . 3 ⊢ (B ∈ A → B ⊆ ∪A)
3	2	adantl 452	. 2 ⊢ ((∪A ⊆ B ∧ B ∈ A) → B ⊆ ∪A)
4	1, 3	eqssd 3290	1 ⊢ ((∪A ⊆ B ∧ B ∈ A) → ∪A = B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  elpwuni  4054