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Mirrors > Home > MPE Home > Th. List > elelpwi | Structured version Visualization version GIF version |
Description: If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
elelpwi | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4117 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶) | |
2 | 1 | sseld 3567 | . 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
3 | 2 | impcom 445 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 𝒫 cpw 4108 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 |
This theorem is referenced by: unipw 4845 axdc2lem 9153 axdc3lem4 9158 homarel 16509 txdis 21245 uhgredgrnv 25804 fpwrelmap 28896 insiga 29527 measinblem 29610 ddemeas 29626 imambfm 29651 totprobd 29815 dstrvprob 29860 ballotlem2 29877 scmsuppss 41947 lincvalsc0 42004 linc0scn0 42006 lincdifsn 42007 linc1 42008 lincsum 42012 lincscm 42013 lcoss 42019 lincext3 42039 islindeps2 42066 |
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