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Theorem eldifpw 4158
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 3343 . . . 4 (A 𝒫 BAB)
2 unss1 3089 . . . . 5 (AB → (A𝐶) ⊆ (B𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 V
4 unexg 4128 . . . . . . 7 ((A 𝒫 B 𝐶 V) → (A𝐶) V)
53, 4mpan2 403 . . . . . 6 (A 𝒫 B → (A𝐶) V)
6 elpwg 3342 . . . . . 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 3343 . . . . 5 ((A𝐶) 𝒫 B → (A𝐶) ⊆ B)
1110unssbd 3098 . . . 4 ((A𝐶) 𝒫 B𝐶B)
1211con3i 549 . . 3 𝐶B → ¬ (A𝐶) 𝒫 B)
139, 12anim12i 321 . 2 ((A 𝒫 B ¬ 𝐶B) → ((A𝐶) 𝒫 (B𝐶) ¬ (A𝐶) 𝒫 B))
14 eldif 2904 . 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 1374  Vcvv 2535  cdif 2891  cun 2892  wss 2894  𝒫 cpw 3334
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by: (None)
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