Theorem limiun 43271

Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6446. (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 6446	. 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		uniiun 5062	. 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	1, 2	eqtrdi 2790	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

Syntax hints:   → wi 4   = wceq 1536  ∪ cuni 4911  ∪ ciun 4995  Lim wlim 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-rex 3068  df-uni 4912  df-iun 4997  df-lim 6390

This theorem is referenced by: (None)