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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 4155 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
2 | 1 | unissi 4809 | . . . . 5 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ ∪ 𝒫 𝐴 |
3 | unipw 5308 | . . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
4 | 2, 3 | sseqtri 3951 | . . . 4 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ 𝐴 |
5 | 4 | sseli 3911 | . . 3 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝐴) |
6 | snelpwi 5302 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ 𝒫 𝐴) | |
7 | snfi 8577 | . . . . . . 7 ⊢ {𝑎} ∈ Fin | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ Fin) |
9 | 6, 8 | elind 4121 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin)) |
10 | elssuni 4830 | . . . . 5 ⊢ ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) |
12 | snidg 4559 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑎}) | |
13 | 11, 12 | sseldd 3916 | . . 3 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin)) |
14 | 5, 13 | impbii 212 | . 2 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) ↔ 𝑎 ∈ 𝐴) |
15 | 14 | eqriv 2795 | 1 ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 Fincfn 8492 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-om 7561 df-1o 8085 df-en 8493 df-fin 8496 |
This theorem is referenced by: isacs5lem 17771 acsmapd 17780 acsmap2d 17781 |
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