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Theorem sspwuni 3730
Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
sspwuni (A ⊆ 𝒫 B AB)

Proof of Theorem sspwuni
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . 4 x V
21elpw 3357 . . 3 (x 𝒫 BxB)
32ralbii 2324 . 2 (x A x 𝒫 Bx A xB)
4 dfss3 2929 . 2 (A ⊆ 𝒫 Bx A x 𝒫 B)
5 unissb 3601 . 2 ( ABx A xB)
63, 4, 53bitr4i 201 1 (A ⊆ 𝒫 B AB)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  wral 2300  wss 2911  𝒫 cpw 3351   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572
This theorem is referenced by:  pwssb  3731  elpwuni  3732  rintm  3735  dftr4  3850  iotass  4827  tfrlemibfn  5883  unirnioo  8612
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