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Theorem prelpwi 5064
Description: A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.)
Assertion
Ref Expression
prelpwi ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

Proof of Theorem prelpwi
StepHypRef Expression
1 prelpw 5063 . 2 ((𝐴𝐶𝐵𝐶) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
21ibi 256 1 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  𝒫 cpw 4302  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324
This theorem is referenced by:  inelfi  8491  elss2prb  13481  isdrs2  17160  usgrexmplef  26371  cusgrexilem2  26569  cusgrfilem2  26583  umgr2v2e  26652  vdegp1bi  26664  eupth2lem3lem5  27405  unelsiga  30527  unelldsys  30551  measxun2  30603  saluncl  41058  prelspr  42264  lincvalpr  42735  ldepspr  42790  zlmodzxzldeplem3  42819  zlmodzxzldep  42821  ldepsnlinc  42825
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