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Theorem unixpss 5771
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 5770 . . . . 5 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
21unissi 4879 . . . 4 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
3 unipw 5412 . . . 4 𝒫 𝒫 (𝐴𝐵) = 𝒫 (𝐴𝐵)
42, 3sseqtri 3985 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
54unissi 4879 . 2 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
6 unipw 5412 . 2 𝒫 (𝐴𝐵) = (𝐴𝐵)
75, 6sseqtri 3985 1 (𝐴 × 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3913  wss 3915  𝒫 cpw 4565   cuni 4870   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  relfld  6232  filnetlem3  34881
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