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Theorem prelpwi 3941
 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.)
Assertion
Ref Expression
prelpwi ((A 𝐶 B 𝐶) → {A, B} 𝒫 𝐶)

Proof of Theorem prelpwi
StepHypRef Expression
1 prssi 3513 . 2 ((A 𝐶 B 𝐶) → {A, B} ⊆ 𝐶)
2 elex 2560 . . . 4 (A 𝐶A V)
3 elex 2560 . . . 4 (B 𝐶B V)
4 prexgOLD 3937 . . . 4 ((A V B V) → {A, B} V)
52, 3, 4syl2an 273 . . 3 ((A 𝐶 B 𝐶) → {A, B} V)
6 elpwg 3359 . . 3 ({A, B} V → ({A, B} 𝒫 𝐶 ↔ {A, B} ⊆ 𝐶))
75, 6syl 14 . 2 ((A 𝐶 B 𝐶) → ({A, B} 𝒫 𝐶 ↔ {A, B} ⊆ 𝐶))
81, 7mpbird 156 1 ((A 𝐶 B 𝐶) → {A, B} 𝒫 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  Vcvv 2551   ⊆ 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935 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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