Theorem unissel 4662

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 470	. 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 ⊆ 𝐵)
2		elssuni 4661	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴)
3	2	adantl 469	. 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴)
4	1, 3	eqssd 3815	1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2156   ⊆ wss 3769  ∪ cuni 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-in 3776  df-ss 3783  df-uni 4631

This theorem is referenced by:  elpwuni  4808  mretopd  21110  toponmre  21111  neiptopuni  21148  filunibas  21898  unidmvol  23522  unicls  30274  carsguni  30695