Theorem limiun 44126

Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6420. (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 6420	. 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		uniiun 5017	. 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	1, 2	eqtrdi 2811	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∪ cuni 4867  ∪ ciun 4951  Lim wlim 6358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-rex 3087  df-uni 4868  df-iun 4953  df-lim 6362

This theorem is used by: (None)