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Theorem eldifpw 4174
Description: Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
Hypothesis
Ref Expression
eldifpw.1 𝐶 V
Assertion
Ref Expression
eldifpw ((A 𝒫 B ¬ 𝐶B) → (A𝐶) (𝒫 (B𝐶) ∖ 𝒫 B))

Proof of Theorem eldifpw
StepHypRef Expression
1 elpwi 3360 . . . 4 (A 𝒫 BAB)
2 unss1 3106 . . . . 5 (AB → (A𝐶) ⊆ (B𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 V
4 unexg 4144 . . . . . . 7 ((A 𝒫 B 𝐶 V) → (A𝐶) V)
53, 4mpan2 401 . . . . . 6 (A 𝒫 B → (A𝐶) V)
6 elpwg 3359 . . . . . 6 ((A𝐶) V → ((A𝐶) 𝒫 (B𝐶) ↔ (A𝐶) ⊆ (B𝐶)))
75, 6syl 14 . . . . 5 (A 𝒫 B → ((A𝐶) 𝒫 (B𝐶) ↔ (A𝐶) ⊆ (B𝐶)))
82, 7syl5ibr 145 . . . 4 (A 𝒫 B → (AB → (A𝐶) 𝒫 (B𝐶)))
91, 8mpd 13 . . 3 (A 𝒫 B → (A𝐶) 𝒫 (B𝐶))
10 elpwi 3360 . . . . 5 ((A𝐶) 𝒫 B → (A𝐶) ⊆ B)
1110unssbd 3115 . . . 4 ((A𝐶) 𝒫 B𝐶B)
1211con3i 561 . . 3 𝐶B → ¬ (A𝐶) 𝒫 B)
139, 12anim12i 321 . 2 ((A 𝒫 B ¬ 𝐶B) → ((A𝐶) 𝒫 (B𝐶) ¬ (A𝐶) 𝒫 B))
14 eldif 2921 . 2 ((A𝐶) (𝒫 (B𝐶) ∖ 𝒫 B) ↔ ((A𝐶) 𝒫 (B𝐶) ¬ (A𝐶) 𝒫 B))
1513, 14sylibr 137 1 ((A 𝒫 B ¬ 𝐶B) → (A𝐶) (𝒫 (B𝐶) ∖ 𝒫 B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wcel 1390  Vcvv 2551  cdif 2908  cun 2909  wss 2911  𝒫 cpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by: (None)
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