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Theorem fiunicl 45072
Description: If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
fiunicl.1 ((𝜑𝑥𝐴𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
fiunicl.2 (𝜑𝐴 ∈ Fin)
fiunicl.3 (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
fiunicl (𝜑 𝐴𝐴)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦

Proof of Theorem fiunicl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 uniiun 5058 . 2 𝐴 = 𝑧𝐴 𝑧
2 nfv 1914 . . 3 𝑧𝜑
3 simpr 484 . . 3 ((𝜑𝑧𝐴) → 𝑧𝐴)
4 fiunicl.1 . . 3 ((𝜑𝑥𝐴𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)
5 fiunicl.2 . . 3 (𝜑𝐴 ∈ Fin)
6 fiunicl.3 . . 3 (𝜑𝐴 ≠ ∅)
72, 3, 4, 5, 6fiiuncl 45070 . 2 (𝜑 𝑧𝐴 𝑧𝐴)
81, 7eqeltrid 2845 1 (𝜑 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  wne 2940  cun 3949  c0 4333   cuni 4907   ciun 4991  Fincfn 8985
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-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989
This theorem is referenced by: (None)
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