ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sspwuni GIF version

Theorem sspwuni 3989
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 2755 . . . 4 𝑥 ∈ V
21elpw 3599 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
32ralbii 2496 . 2 (∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 dfss3 3160 . 2 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵)
5 unissb 3857 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
63, 4, 53bitr4i 212 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2160  wral 2468  wss 3144  𝒫 cpw 3593   cuni 3827
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-pw 3595  df-uni 3828
This theorem is referenced by:  pwssb  3990  elpwpw  3991  elpwuni  3994  rintm  3997  dftr4  4124  iotass  5216  tfrlemibfn  6357  tfr1onlembfn  6373  tfrcllembfn  6386  uniixp  6751  fipwssg  7012  unirnioo  10009  restid  12766  lssintclm  13725  topgele  14014  topontopn  14022  unitg  14047  epttop  14075  resttopon  14156  txuni2  14241  txdis  14262  unirnblps  14407  unirnbl  14408
  Copyright terms: Public domain W3C validator