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Theorem fiunicl 44311
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 5054 . 2 𝐴 = 𝑧𝐴 𝑧
2 nfv 1909 . . 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 44309 . 2 (𝜑 𝑧𝐴 𝑧𝐴)
81, 7eqeltrid 2831 1 (𝜑 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  wne 2934  cun 3941  c0 4317   cuni 4902   ciun 4990  Fincfn 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-en 8939  df-fin 8942
This theorem is referenced by: (None)
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