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Theorem limiun 43295
Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6445. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
limiun (Lim 𝐴𝐴 = 𝑥𝐴 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem limiun
StepHypRef Expression
1 limuni 6445 . 2 (Lim 𝐴𝐴 = 𝐴)
2 uniiun 5058 . 2 𝐴 = 𝑥𝐴 𝑥
31, 2eqtrdi 2793 1 (Lim 𝐴𝐴 = 𝑥𝐴 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540   cuni 4907   ciun 4991  Lim wlim 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-rex 3071  df-uni 4908  df-iun 4993  df-lim 6389
This theorem is referenced by: (None)
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