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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 4612 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶) | |
2 | 1 | sseld 3994 | . 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
3 | 2 | impcom 407 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ss 3980 df-pw 4607 |
This theorem is referenced by: unipw 5461 axdc2lem 10486 axdc3lem4 10491 homarel 18090 txdis 23656 uhgredgrnv 29162 fpwrelmap 32751 insiga 34118 measinblem 34201 ddemeas 34217 imambfm 34244 totprobd 34408 dstrvprob 34453 ballotlem2 34470 requad2 47548 scmsuppss 48216 lincvalsc0 48267 linc0scn0 48269 lincdifsn 48270 linc1 48271 lincsum 48275 lincscm 48276 lcoss 48282 lincext3 48302 islindeps2 48329 itscnhlinecirc02p 48635 |
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