Theorem prelpwi 5393

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 5392	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
2	1	ibi 268	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  𝒫 cpw 4536  {cpr 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-pw 4538  df-sn 4563  df-pr 4565

This theorem is referenced by:  inelfi  9328  elss2prb  14448  isdrs2  18270  usgrexmplef  29353  cusgrexilem2  29536  cusgrfilem2  29550  umgr2v2e  29619  vdegp1bi  29631  eupth2lem3lem5  30327  unelsiga  34325  inelpisys  34345  unelldsys  34349  measxun2  34401  saluncl  46767  prelspr  47968  prpair  47983  prproropf1olem1  47985  paireqne  47993  prprelprb  47999  isgrtri  48441  stgr1  48459  gpgprismgr4cycllem3  48595  lincvalpr  48916  ldepspr  48971  zlmodzxzldeplem3  49000  zlmodzxzldep  49002  ldepsnlinc  49006