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Theorem elpwuni 3741
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 3739 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissel 3609 . . . 4 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
32expcom 109 . . 3 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
4 eqimss 2997 . . 3 ( 𝐴 = 𝐵 𝐴𝐵)
53, 4impbid1 130 . 2 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
61, 5syl5bb 181 1 (𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  wss 2917  𝒫 cpw 3359   cuni 3580
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581
This theorem is referenced by: (None)
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