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Theorem prelpwi 3977
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 3549 . 2 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
2 elex 2583 . . . 4 (𝐴𝐶𝐴 ∈ V)
3 elex 2583 . . . 4 (𝐵𝐶𝐵 ∈ V)
4 prexgOLD 3973 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
52, 3, 4syl2an 277 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ V)
6 elpwg 3394 . . 3 ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
75, 6syl 14 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
81, 7mpbird 160 1 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  Vcvv 2574  wss 2944  𝒫 cpw 3386  {cpr 3403
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pr 3971
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 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409
This theorem is referenced by: (None)
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