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Theorem sspwuni 3811
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 2622 . . . 4 𝑥 ∈ V
21elpw 3433 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
32ralbii 2384 . 2 (∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 dfss3 3015 . 2 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵)
5 unissb 3681 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
63, 4, 53bitr4i 210 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1438  wral 2359  wss 2999  𝒫 cpw 3427   cuni 3651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-pw 3429  df-uni 3652
This theorem is referenced by:  pwssb  3812  elpwpw  3813  elpwuni  3816  rintm  3819  dftr4  3939  iotass  4992  tfrlemibfn  6085  tfr1onlembfn  6101  tfrcllembfn  6114  unirnioo  9381  topgele  11508
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