Theorem prelpwi 5410

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 5409	. 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 2109  𝒫 cpw 4566  {cpr 4594

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595

This theorem is referenced by:  inelfi  9376  elss2prb  14460  isdrs2  18274  usgrexmplef  29193  cusgrexilem2  29376  cusgrfilem2  29391  umgr2v2e  29460  vdegp1bi  29472  eupth2lem3lem5  30168  unelsiga  34131  inelpisys  34151  unelldsys  34155  measxun2  34207  saluncl  46322  prelspr  47491  prpair  47506  prproropf1olem1  47508  paireqne  47516  prprelprb  47522  isgrtri  47946  stgr1  47964  gpgprismgr4cycllem3  48091  lincvalpr  48411  ldepspr  48466  zlmodzxzldeplem3  48495  zlmodzxzldep  48497  ldepsnlinc  48501