Theorem fiunicl 42615

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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴)
fiunicl.2	⊢ (𝜑 → 𝐴 ∈ Fin)
fiunicl.3	⊢ (𝜑 → 𝐴 ≠ ∅)

Assertion

Ref	Expression
fiunicl	⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦

Proof of Theorem fiunicl

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniiun 4988	. 2 ⊢ ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 𝑧
2		nfv 1917	. . 3 ⊢ Ⅎ𝑧𝜑
3		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
4		fiunicl.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴)
5		fiunicl.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fiunicl.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 3, 4, 5, 6	fiiuncl 42613	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝐴 𝑧 ∈ 𝐴)
8	1, 7	eqeltrid 2843	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2106   ≠ wne 2943   ∪ cun 3885  ∅c0 4256  ∪ cuni 4839  ∪ ciun 4924  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-en 8734  df-fin 8737

This theorem is referenced by: (None)