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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 5447 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) | |
2 | 1 | ibi 267 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 𝒫 cpw 4603 {cpr 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 |
This theorem is referenced by: inelfi 9413 elss2prb 14448 isdrs2 18259 usgrexmplef 28516 cusgrexilem2 28699 cusgrfilem2 28713 umgr2v2e 28782 vdegp1bi 28794 eupth2lem3lem5 29485 unelsiga 33132 inelpisys 33152 unelldsys 33156 measxun2 33208 saluncl 45033 prelspr 46154 prpair 46169 prproropf1olem1 46171 paireqne 46179 prprelprb 46185 lincvalpr 47099 ldepspr 47154 zlmodzxzldeplem3 47183 zlmodzxzldep 47185 ldepsnlinc 47189 |
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