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Theorem elpwuni 3938
 Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
elpwuni (𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))

Proof of Theorem elpwuni
StepHypRef Expression
1 sspwuni 3933 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissel 3801 . . . 4 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
32expcom 115 . . 3 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
4 eqimss 3182 . . 3 ( 𝐴 = 𝐵 𝐴𝐵)
53, 4impbid1 141 . 2 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
61, 5syl5bb 191 1 (𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335   ∈ wcel 2128   ⊆ wss 3102  𝒫 cpw 3543  ∪ cuni 3772 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-uni 3773 This theorem is referenced by: (None)
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