MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prsspw Structured version   Visualization version   GIF version

Theorem prsspw 4374
Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
Hypotheses
Ref Expression
prsspw.1 𝐴 ∈ V
prsspw.2 𝐵 ∈ V
Assertion
Ref Expression
prsspw ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))

Proof of Theorem prsspw
StepHypRef Expression
1 prsspw.1 . 2 𝐴 ∈ V
2 prsspw.2 . 2 𝐵 ∈ V
3 prsspwg 4353 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
41, 2, 3mp2an 708 1 ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1989  Vcvv 3198  wss 3572  𝒫 cpw 4156  {cpr 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-un 3577  df-in 3579  df-ss 3586  df-pw 4158  df-sn 4176  df-pr 4178
This theorem is referenced by:  altxpsspw  32068
  Copyright terms: Public domain W3C validator