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Theorem fiunicl 38718
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 4539 . 2 𝐴 = 𝑧𝐴 𝑧
2 nfv 1840 . . 3 𝑧𝜑
3 simpr 477 . . 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 38716 . 2 (𝜑 𝑧𝐴 𝑧𝐴)
81, 7syl5eqel 2702 1 (𝜑 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1987  wne 2790  cun 3553  c0 3891   cuni 4402   ciun 4485  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-fin 7903
This theorem is referenced by: (None)
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