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Theorem unixpss 5816
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 5815 . . . . 5 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
21unissi 4921 . . . 4 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
3 unipw 5456 . . . 4 𝒫 𝒫 (𝐴𝐵) = 𝒫 (𝐴𝐵)
42, 3sseqtri 4018 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
54unissi 4921 . 2 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
6 unipw 5456 . 2 𝒫 (𝐴𝐵) = (𝐴𝐵)
75, 6sseqtri 4018 1 (𝐴 × 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3947  wss 3949  𝒫 cpw 4606   cuni 4912   × cxp 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by:  relfld  6284  filnetlem3  35897
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