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Theorem prsspwg 4748
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 4744 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶))
2 elpwg 4543 . . 3 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐶𝐴𝐶))
3 elpwg 4543 . . 3 (𝐵𝑊 → (𝐵 ∈ 𝒫 𝐶𝐵𝐶))
42, 3bi2anan9 637 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ (𝐴𝐶𝐵𝐶)))
51, 4bitr3d 283 1 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2107  wss 3934  𝒫 cpw 4537  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-in 3941  df-ss 3950  df-pw 4539  df-sn 4560  df-pr 4562
This theorem is referenced by:  prsspw  4768
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