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Theorem fiunicl 44462
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 5065 . 2 𝐴 = 𝑧𝐴 𝑧
2 nfv 1909 . . 3 𝑧𝜑
3 simpr 483 . . 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 44460 . 2 (𝜑 𝑧𝐴 𝑧𝐴)
81, 7eqeltrid 2833 1 (𝜑 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  wne 2937  cun 3947  c0 4326   cuni 4912   ciun 5000  Fincfn 8970
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-en 8971  df-fin 8974
This theorem is referenced by: (None)
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