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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 3795 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
2 | 1 | unissi 4397 | . . . . 5 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ ∪ 𝒫 𝐴 |
3 | unipw 4845 | . . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
4 | 2, 3 | sseqtri 3600 | . . . 4 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ 𝐴 |
5 | 4 | sseli 3564 | . . 3 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝐴) |
6 | snelpwi 4839 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ 𝒫 𝐴) | |
7 | snfi 7923 | . . . . . . 7 ⊢ {𝑎} ∈ Fin | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ Fin) |
9 | 6, 8 | elind 3760 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin)) |
10 | elssuni 4403 | . . . . 5 ⊢ ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) |
12 | snidg 4153 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑎}) | |
13 | 11, 12 | sseldd 3569 | . . 3 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin)) |
14 | 5, 13 | impbii 198 | . 2 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) ↔ 𝑎 ∈ 𝐴) |
15 | 14 | eqriv 2607 | 1 ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-om 6958 df-1o 7447 df-en 7842 df-fin 7845 |
This theorem is referenced by: isacs5lem 16992 acsmapd 17001 acsmap2d 17002 |
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