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Theorem unixpss 5666
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 5665 . . . . 5 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
21unissi 4830 . . . 4 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
3 unipw 5326 . . . 4 𝒫 𝒫 (𝐴𝐵) = 𝒫 (𝐴𝐵)
42, 3sseqtri 3987 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
54unissi 4830 . 2 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
6 unipw 5326 . 2 𝒫 (𝐴𝐵) = (𝐴𝐵)
75, 6sseqtri 3987 1 (𝐴 × 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3916  wss 3918  𝒫 cpw 4520   cuni 4821   × cxp 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-opab 5112  df-xp 5544  df-rel 5545
This theorem is referenced by:  relfld  6109  filnetlem3  33748
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