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Theorem sspwuni 3897
 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 2689 . . . 4 𝑥 ∈ V
21elpw 3516 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
32ralbii 2441 . 2 (∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 dfss3 3087 . 2 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵)
5 unissb 3766 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
63, 4, 53bitr4i 211 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737 This theorem is referenced by:  pwssb  3898  elpwpw  3899  elpwuni  3902  rintm  3905  dftr4  4031  iotass  5105  tfrlemibfn  6225  tfr1onlembfn  6241  tfrcllembfn  6254  uniixp  6615  fipwssg  6867  unirnioo  9768  restid  12145  topgele  12210  topontopn  12218  unitg  12245  epttop  12273  resttopon  12354  txuni2  12439  txdis  12460  unirnblps  12605  unirnbl  12606
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