Theorem limiun 43710

Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6385. (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 6385	. 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		uniiun 5001	. 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	1, 2	eqtrdi 2787	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

Syntax hints:   → wi 4   = wceq 1542  ∪ cuni 4850  ∪ ciun 4933  Lim wlim 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-rex 3062  df-uni 4851  df-iun 4935  df-lim 6328

This theorem is referenced by: (None)