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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 5451 | . 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 2108 𝒫 cpw 4600 {cpr 4628 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: inelfi 9458 elss2prb 14527 isdrs2 18352 usgrexmplef 29276 cusgrexilem2 29459 cusgrfilem2 29474 umgr2v2e 29543 vdegp1bi 29555 eupth2lem3lem5 30251 unelsiga 34135 inelpisys 34155 unelldsys 34159 measxun2 34211 saluncl 46332 prelspr 47473 prpair 47488 prproropf1olem1 47490 paireqne 47498 prprelprb 47504 isgrtri 47910 stgr1 47928 lincvalpr 48335 ldepspr 48390 zlmodzxzldeplem3 48419 zlmodzxzldep 48421 ldepsnlinc 48425 |
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