Theorem limiun 43727

Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6372. (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 6372	. 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		uniiun 4988	. 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	1, 2	eqtrdi 2790	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

Syntax hints:   → wi 4   = wceq 1547  ∪ cuni 4838  ∪ ciun 4921  Lim wlim 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-rex 3064  df-uni 4839  df-iun 4923  df-lim 6315

This theorem is referenced by: (None)