Theorem prelpwi 5331

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.)

Assertion

Ref	Expression
prelpwi	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

Proof of Theorem prelpwi

Step	Hyp	Ref	Expression
1		prelpw 5330	. 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)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  𝒫 cpw 4538  {cpr 4562

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563

This theorem is referenced by:  inelfi  8876  elss2prb  13839  isdrs2  17543  usgrexmplef  27035  cusgrexilem2  27218  cusgrfilem2  27232  umgr2v2e  27301  vdegp1bi  27313  eupth2lem3lem5  28005  unelsiga  31388  unelldsys  31412  measxun2  31464  saluncl  42596  prelspr  43642  prpair  43657  prproropf1olem1  43659  paireqne  43667  prprelprb  43673  lincvalpr  44467  ldepspr  44522  zlmodzxzldeplem3  44551  zlmodzxzldep  44553  ldepsnlinc  44557