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Theorem elpwuni 3988
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 3983 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissel 3850 . . . 4 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
32expcom 116 . . 3 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
4 eqimss 3221 . . 3 ( 𝐴 = 𝐵 𝐴𝐵)
53, 4impbid1 142 . 2 (𝐵𝐴 → ( 𝐴𝐵 𝐴 = 𝐵))
61, 5bitrid 192 1 (𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  wcel 2158  wss 3141  𝒫 cpw 3587   cuni 3821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589  df-uni 3822
This theorem is referenced by: (None)
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