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Theorem sspwuni 3709
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 2534 . . . 4 x V
21elpw 3336 . . 3 (x 𝒫 BxB)
32ralbii 2304 . 2 (x A x 𝒫 Bx A xB)
4 dfss3 2908 . 2 (A ⊆ 𝒫 Bx A x 𝒫 B)
5 unissb 3580 . 2 ( ABx A xB)
63, 4, 53bitr4i 201 1 (A ⊆ 𝒫 B AB)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1370  wral 2280  wss 2890  𝒫 cpw 3330   cuni 3550
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332  df-uni 3551
This theorem is referenced by:  pwssb  3710  elpwuni  3711  rintm  3714  dftr4  3829  iotass  4807  tfrlemibfn  5859
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