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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 4237 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
| 2 | 1 | unissi 4916 | . . . . 5 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ ∪ 𝒫 𝐴 |
| 3 | unipw 5455 | . . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
| 4 | 2, 3 | sseqtri 4032 | . . . 4 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ 𝐴 |
| 5 | 4 | sseli 3979 | . . 3 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝐴) |
| 6 | snelpwi 5448 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ 𝒫 𝐴) | |
| 7 | snfi 9083 | . . . . . . 7 ⊢ {𝑎} ∈ Fin | |
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ Fin) |
| 9 | 6, 8 | elind 4200 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin)) |
| 10 | elssuni 4937 | . . . . 5 ⊢ ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) | |
| 11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) |
| 12 | snidg 4660 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑎}) | |
| 13 | 11, 12 | sseldd 3984 | . . 3 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin)) |
| 14 | 5, 13 | impbii 209 | . 2 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) ↔ 𝑎 ∈ 𝐴) |
| 15 | 14 | eqriv 2734 | 1 ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 |
| This theorem is referenced by: isacs5lem 18590 acsmapd 18599 acsmap2d 18600 |
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