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Mirrors > Home > MPE Home > Th. List > prelpwi | Structured version Visualization version GIF version |
Description: A pair of two sets belongs to the power class of a class containing those two sets. (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 5341 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) | |
2 | 1 | ibi 269 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 𝒫 cpw 4541 {cpr 4571 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-sn 4570 df-pr 4572 |
This theorem is referenced by: inelfi 8884 elss2prb 13848 isdrs2 17551 usgrexmplef 27043 cusgrexilem2 27226 cusgrfilem2 27240 umgr2v2e 27309 vdegp1bi 27321 eupth2lem3lem5 28013 unelsiga 31395 unelldsys 31419 measxun2 31471 saluncl 42609 prelspr 43655 prpair 43670 prproropf1olem1 43672 paireqne 43680 prprelprb 43686 lincvalpr 44480 ldepspr 44535 zlmodzxzldeplem3 44564 zlmodzxzldep 44566 ldepsnlinc 44570 |
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