Theorem prelpwi 5394

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 5393	. 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 4553  {cpr 4581

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-pw 4555  df-sn 4580  df-pr 4582

This theorem is referenced by:  inelfi  9327  elss2prb  14413  isdrs2  18230  usgrexmplef  29222  cusgrexilem2  29405  cusgrfilem2  29420  umgr2v2e  29489  vdegp1bi  29501  eupth2lem3lem5  30194  unelsiga  34100  inelpisys  34120  unelldsys  34124  measxun2  34176  saluncl  46299  prelspr  47471  prpair  47486  prproropf1olem1  47488  paireqne  47496  prprelprb  47502  isgrtri  47928  stgr1  47946  gpgprismgr4cycllem3  48082  lincvalpr  48404  ldepspr  48459  zlmodzxzldeplem3  48488  zlmodzxzldep  48490  ldepsnlinc  48494