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Theorem elpwuni 4804
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 4799 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissel 4658 . . . 4 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
32expcom 400 . . 3 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
4 eqimss 3851 . . 3 ( 𝐴 = 𝐵 𝐴𝐵)
53, 4impbid1 216 . 2 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
61, 5syl5bb 274 1 (𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wcel 2158  wss 3766  𝒫 cpw 4348   cuni 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-v 3392  df-in 3773  df-ss 3780  df-pw 4350  df-uni 4627
This theorem is referenced by:  mreuni  16461  ustuni  22239  utopbas  22248  issgon  30508  br2base  30653
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