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Theorem prsspwg 3551
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 3549 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶))
2 elpwg 3395 . . 3 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐶𝐴𝐶))
3 elpwg 3395 . . 3 (𝐵𝑊 → (𝐵 ∈ 𝒫 𝐶𝐵𝐶))
42, 3bi2anan9 548 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ (𝐴𝐶𝐵𝐶)))
51, 4bitr3d 183 1 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  wss 2945  𝒫 cpw 3387  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410
This theorem is referenced by: (None)
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