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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 4230 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
2 | 1 | unissi 4922 | . . . . 5 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ ∪ 𝒫 𝐴 |
3 | unipw 5456 | . . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
4 | 2, 3 | sseqtri 4016 | . . . 4 ⊢ ∪ (𝒫 𝐴 ∩ Fin) ⊆ 𝐴 |
5 | 4 | sseli 3975 | . . 3 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝐴) |
6 | snelpwi 5449 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ 𝒫 𝐴) | |
7 | snfi 9081 | . . . . . . 7 ⊢ {𝑎} ∈ Fin | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ Fin) |
9 | 6, 8 | elind 4195 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin)) |
10 | elssuni 4945 | . . . . 5 ⊢ ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ ∪ (𝒫 𝐴 ∩ Fin)) |
12 | snidg 4667 | . . . 4 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑎}) | |
13 | 11, 12 | sseldd 3980 | . . 3 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin)) |
14 | 5, 13 | impbii 208 | . 2 ⊢ (𝑎 ∈ ∪ (𝒫 𝐴 ∩ Fin) ↔ 𝑎 ∈ 𝐴) |
15 | 14 | eqriv 2723 | 1 ⊢ ∪ (𝒫 𝐴 ∩ Fin) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4607 {csn 4633 ∪ cuni 4913 Fincfn 8974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-om 7877 df-1o 8496 df-en 8975 df-fin 8978 |
This theorem is referenced by: isacs5lem 18570 acsmapd 18579 acsmap2d 18580 |
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