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Theorem elpwuni 3866
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 3861 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissel 3729 . . . 4 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
32expcom 115 . . 3 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
4 eqimss 3115 . . 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 1312  wcel 1461  wss 3035  𝒫 cpw 3474   cuni 3700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-uni 3701
This theorem is referenced by: (None)
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