Theorem prelpwi 5422

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.)

Assertion

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

Proof of Theorem prelpwi

Step	Hyp	Ref	Expression
1		prelpw 5421	. 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)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  𝒫 cpw 4575  {cpr 4603

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-pw 4577  df-sn 4602  df-pr 4604

This theorem is referenced by:  inelfi  9430  elss2prb  14506  isdrs2  18318  usgrexmplef  29238  cusgrexilem2  29421  cusgrfilem2  29436  umgr2v2e  29505  vdegp1bi  29517  eupth2lem3lem5  30213  unelsiga  34165  inelpisys  34185  unelldsys  34189  measxun2  34241  saluncl  46346  prelspr  47500  prpair  47515  prproropf1olem1  47517  paireqne  47525  prprelprb  47531  isgrtri  47955  stgr1  47973  gpgprismgr4cycllem3  48096  lincvalpr  48394  ldepspr  48449  zlmodzxzldeplem3  48478  zlmodzxzldep  48480  ldepsnlinc  48484