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Mirrors > Home > MPE Home > Th. List > prelpwi | Structured version Visualization version GIF version |
Description: If two sets are members of a class, then the unordered pair of those two sets is a member of the powerclass of that class. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.) |
Ref | Expression |
---|---|
prelpwi | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prelpw 5445 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) | |
2 | 1 | ibi 266 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2104 𝒫 cpw 4601 {cpr 4629 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-pw 4603 df-sn 4628 df-pr 4630 |
This theorem is referenced by: inelfi 9415 elss2prb 14452 isdrs2 18263 usgrexmplef 28783 cusgrexilem2 28966 cusgrfilem2 28980 umgr2v2e 29049 vdegp1bi 29061 eupth2lem3lem5 29752 unelsiga 33430 inelpisys 33450 unelldsys 33454 measxun2 33506 saluncl 45331 prelspr 46452 prpair 46467 prproropf1olem1 46469 paireqne 46477 prprelprb 46483 lincvalpr 47186 ldepspr 47241 zlmodzxzldeplem3 47270 zlmodzxzldep 47272 ldepsnlinc 47276 |
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