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Theorem prsspw 3687
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.)
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 . . 3 𝐴 ∈ V
2 prsspw.2 . . 3 𝐵 ∈ V
31, 2prss 3671 . 2 ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶)
41elpw 3511 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
52elpw 3511 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
64, 5anbi12i 455 . 2 ((𝐴 ∈ 𝒫 𝐶𝐵 ∈ 𝒫 𝐶) ↔ (𝐴𝐶𝐵𝐶))
73, 6bitr3i 185 1 ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1480  Vcvv 2681  wss 3066  𝒫 cpw 3505  {cpr 3523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529
This theorem is referenced by: (None)
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