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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 4506 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶) | |
2 | 1 | sseld 3914 | . 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
3 | 2 | impcom 411 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 𝒫 cpw 4497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 |
This theorem is referenced by: unipw 5308 axdc2lem 9859 axdc3lem4 9864 homarel 17288 txdis 22237 uhgredgrnv 26923 fpwrelmap 30495 insiga 31506 measinblem 31589 ddemeas 31605 imambfm 31630 totprobd 31794 dstrvprob 31839 ballotlem2 31856 requad2 44141 scmsuppss 44774 lincvalsc0 44830 linc0scn0 44832 lincdifsn 44833 linc1 44834 lincsum 44838 lincscm 44839 lcoss 44845 lincext3 44865 islindeps2 44892 itscnhlinecirc02p 45199 |
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