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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 4548 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶) | |
2 | 1 | sseld 3966 | . 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
3 | 2 | impcom 410 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-pw 4541 |
This theorem is referenced by: unipw 5343 axdc2lem 9870 axdc3lem4 9875 homarel 17296 txdis 22240 uhgredgrnv 26915 fpwrelmap 30469 insiga 31396 measinblem 31479 ddemeas 31495 imambfm 31520 totprobd 31684 dstrvprob 31729 ballotlem2 31746 requad2 43808 scmsuppss 44440 lincvalsc0 44496 linc0scn0 44498 lincdifsn 44499 linc1 44500 lincsum 44504 lincscm 44505 lcoss 44511 lincext3 44531 islindeps2 44558 itscnhlinecirc02p 44792 |
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