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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 5065 | . 2 ⊢ ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 𝑧 | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | simpr 483 | . . 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 44460 | . 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝐴 𝑧 ∈ 𝐴) |
8 | 1, 7 | eqeltrid 2833 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) |
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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