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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fiunicl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fiunicl.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴) |
fiunicl.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fiunicl.3 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
fiunicl | ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) |
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) |
Colors of variables: wff setvar class |
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) |
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