MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unixpss Structured version   Visualization version   GIF version

Theorem unixpss 5820
Description: The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
unixpss (𝐴 × 𝐵) ⊆ (𝐴𝐵)

Proof of Theorem unixpss
StepHypRef Expression
1 xpsspw 5819 . . . . 5 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
21unissi 4916 . . . 4 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
3 unipw 5455 . . . 4 𝒫 𝒫 (𝐴𝐵) = 𝒫 (𝐴𝐵)
42, 3sseqtri 4032 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
54unissi 4916 . 2 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
6 unipw 5455 . 2 𝒫 (𝐴𝐵) = (𝐴𝐵)
75, 6sseqtri 4032 1 (𝐴 × 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3949  wss 3951  𝒫 cpw 4600   cuni 4907   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  relfld  6295  filnetlem3  36381
  Copyright terms: Public domain W3C validator