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Theorem prelpwi 4147
 Description: A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
Assertion
Ref Expression
prelpwi ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

Proof of Theorem prelpwi
StepHypRef Expression
1 prssi 3688 . 2 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
2 prexg 4144 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ V)
3 elpwg 3525 . . 3 ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
42, 3syl 14 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
51, 4mpbird 166 1 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2112  Vcvv 2691   ⊆ wss 3078  𝒫 cpw 3517  {cpr 3535 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541 This theorem is referenced by: (None)
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