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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 5457 | . 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 395 ∈ wcel 2106 𝒫 cpw 4605 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 |
This theorem is referenced by: inelfi 9456 elss2prb 14524 isdrs2 18364 usgrexmplef 29291 cusgrexilem2 29474 cusgrfilem2 29489 umgr2v2e 29558 vdegp1bi 29570 eupth2lem3lem5 30261 unelsiga 34115 inelpisys 34135 unelldsys 34139 measxun2 34191 saluncl 46273 prelspr 47411 prpair 47426 prproropf1olem1 47428 paireqne 47436 prprelprb 47442 isgrtri 47848 stgr1 47864 lincvalpr 48264 ldepspr 48319 zlmodzxzldeplem3 48348 zlmodzxzldep 48350 ldepsnlinc 48354 |
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