Theorem prelpwi 5390

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 5389	. 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 2113  𝒫 cpw 4549  {cpr 4577

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-pw 4551  df-sn 4576  df-pr 4578

This theorem is referenced by:  inelfi  9309  elss2prb  14397  isdrs2  18214  usgrexmplef  29239  cusgrexilem2  29422  cusgrfilem2  29437  umgr2v2e  29506  vdegp1bi  29518  eupth2lem3lem5  30214  unelsiga  34168  inelpisys  34188  unelldsys  34192  measxun2  34244  saluncl  46439  prelspr  47610  prpair  47625  prproropf1olem1  47627  paireqne  47635  prprelprb  47641  isgrtri  48067  stgr1  48085  gpgprismgr4cycllem3  48221  lincvalpr  48543  ldepspr  48598  zlmodzxzldeplem3  48627  zlmodzxzldep  48629  ldepsnlinc  48633