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 Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
Assertion
Ref Expression
prsspwg ((A 𝑉 B 𝑊) → ({A, B} ⊆ 𝒫 𝐶 ↔ (A𝐶 B𝐶)))

StepHypRef Expression
1 prssg 3512 . 2 ((A 𝑉 B 𝑊) → ((A 𝒫 𝐶 B 𝒫 𝐶) ↔ {A, B} ⊆ 𝒫 𝐶))
2 elpwg 3359 . . 3 (A 𝑉 → (A 𝒫 𝐶A𝐶))
3 elpwg 3359 . . 3 (B 𝑊 → (B 𝒫 𝐶B𝐶))
42, 3bi2anan9 538 . 2 ((A 𝑉 B 𝑊) → ((A 𝒫 𝐶 B 𝒫 𝐶) ↔ (A𝐶 B𝐶)))
51, 4bitr3d 179 1 ((A 𝑉 B 𝑊) → ({A, B} ⊆ 𝒫 𝐶 ↔ (A𝐶 B𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351  {cpr 3368 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374 This theorem is referenced by: (None)
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