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Theorem elpwuni 3715
 Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
elpwuni (B A → (A ⊆ 𝒫 B A = B))

Proof of Theorem elpwuni
StepHypRef Expression
1 sspwuni 3713 . 2 (A ⊆ 𝒫 B AB)
2 unissel 3583 . . . 4 (( AB B A) → A = B)
32expcom 109 . . 3 (B A → ( AB A = B))
4 eqimss 2974 . . 3 ( A = B AB)
53, 4impbid1 130 . 2 (B A → ( AB A = B))
61, 5syl5bb 181 1 (B A → (A ⊆ 𝒫 B A = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894  𝒫 cpw 3334  ∪ cuni 3554 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-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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 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-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-uni 3555 This theorem is referenced by: (None)
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