Theorem limiun 43859

Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6408. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
limiun	⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem limiun

Step	Hyp	Ref	Expression
1		limuni 6408	. 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		uniiun 5016	. 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	1, 2	eqtrdi 2813	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

Syntax hints:   → wi 4   = wceq 1560  ∪ cuni 4865  ∪ ciun 4949  Lim wlim 6347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-rex 3087  df-uni 4866  df-iun 4951  df-lim 6351

This theorem is referenced by: (None)