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Mirrors > Home > MPE Home > Th. List > unifi | Structured version Visualization version GIF version |
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
unifi | ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3970 | . 2 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) | |
2 | uniiun 5061 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
3 | iunfi 9339 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ Fin) | |
4 | 2, 3 | eqeltrid 2837 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) → ∪ 𝐴 ∈ Fin) |
5 | 1, 4 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 ∪ ciun 4997 Fincfn 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7855 df-en 8939 df-fin 8942 |
This theorem is referenced by: unifi2 9341 unirnffid 9343 incexc 15782 incexc2 15783 discmp 22901 tsmsxplem1 23656 fpwrelmapffslem 31952 heiborlem1 36674 |
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