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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 4607 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶) | |
| 2 | 1 | sseld 3982 | . 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 2108 𝒫 cpw 4600 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: unipw 5455 axdc2lem 10488 axdc3lem4 10493 homarel 18081 txdis 23640 uhgredgrnv 29147 fpwrelmap 32744 insiga 34138 measinblem 34221 ddemeas 34237 imambfm 34264 totprobd 34428 dstrvprob 34474 ballotlem2 34491 requad2 47610 scmsuppss 48287 lincvalsc0 48338 linc0scn0 48340 lincdifsn 48341 linc1 48342 lincsum 48346 lincscm 48347 lcoss 48353 lincext3 48373 islindeps2 48400 itscnhlinecirc02p 48706 |
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