Theorem fiunicl 40786

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 4844	. 2 ⊢ ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 𝑧
2		nfv 1874	. . 3 ⊢ Ⅎ𝑧𝜑
3		simpr 477	. . 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 40784	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝐴 𝑧 ∈ 𝐴)
8	1, 7	syl5eqel 2863	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1069   ∈ wcel 2051   ≠ wne 2960   ∪ cun 3820  ∅c0 4172  ∪ cuni 4708  ∪ ciun 4788  Fincfn 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-om 7395  df-1o 7903  df-er 8087  df-en 8305  df-fin 8308

This theorem is referenced by: (None)