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Theorem prsspwg 4823
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 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Proof of Theorem prsspwg
StepHypRef Expression
1 prssg 4819 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶))
2 elpwg 4603 . . 3 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐶𝐴𝐶))
3 elpwg 4603 . . 3 (𝐵𝑊 → (𝐵 ∈ 𝒫 𝐶𝐵𝐶))
42, 3bi2anan9 638 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ (𝐴𝐶𝐵𝐶)))
51, 4bitr3d 281 1 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wss 3951  𝒫 cpw 4600  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629
This theorem is referenced by:  prsspw  4845
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