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Mirrors > Home > MPE Home > Th. List > unifpw | Structured version Visualization version GIF version |
Description: A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
unifpw | ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4204 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
2 | 1 | unissi 4846 | . . . . 5 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ ∪ 𝒫 𝐴 |
3 | unipw 5342 | . . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
4 | 2, 3 | sseqtri 4002 | . . . 4 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ 𝐴 |
5 | 4 | sseli 3962 | . . 3 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝐴) |
6 | snelpwi 5336 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ 𝒫 𝐴) | |
7 | snfi 8593 | . . . . . . 7 ⊢ {𝑎} ∈ Fin | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ Fin) |
9 | 6, 8 | elind 4170 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin)) |
10 | elssuni 4867 | . . . . 5 ⊢ ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) |
12 | snidg 4598 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑎}) | |
13 | 11, 12 | sseldd 3967 | . . 3 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin)) |
14 | 5, 13 | impbii 211 | . 2 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) ↔ 𝑎 ∈ 𝐴) |
15 | 14 | eqriv 2818 | 1 ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 {csn 4566 ∪ cuni 4837 Fincfn 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-om 7580 df-1o 8101 df-en 8509 df-fin 8512 |
This theorem is referenced by: isacs5lem 17778 acsmapd 17787 acsmap2d 17788 |
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