Theorem prelpwi 5433

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 5432	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
2	1	ibi 270	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  𝒫 cpw 4567  {cpr 4596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-un 3913  df-ss 3925  df-pw 4569  df-sn 4595  df-pr 4597

This theorem is used by:  inelfi  9388  elss2prb  14545  isdrs2  18387  usgrexmplef  29639  cusgrexilem2  29822  cusgrfilem2  29836  umgr2v2e  29905  vdegp1bi  29917  eupth2lem3lem5  30613  unelsiga  34548  inelpisys  34568  unelldsys  34572  measxun2  34624  saluncl  47064  prelspr  48268  prpair  48283  prproropf1olem1  48285  paireqne  48293  prprelprb  48299  isgrtri  48741  stgr1  48759  gpgprismgr4cycllem3  48895  lincvalpr  49231  ldepspr  49286  zlmodzxzldeplem3  49315  zlmodzxzldep  49317  ldepsnlinc  49321