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Theorem unixpss 4451
 Description: The double class union of a cross 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 4450 . . . . 5 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
21unissi 3603 . . . 4 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
3 unipw 3953 . . . 4 𝒫 𝒫 (𝐴𝐵) = 𝒫 (𝐴𝐵)
42, 3sseqtri 2977 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
54unissi 3603 . 2 (𝐴 × 𝐵) ⊆ 𝒫 (𝐴𝐵)
6 unipw 3953 . 2 𝒫 (𝐴𝐵) = (𝐴𝐵)
75, 6sseqtri 2977 1 (𝐴 × 𝐵) ⊆ (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∪ cun 2915   ⊆ wss 2917  𝒫 cpw 3359  ∪ cuni 3580   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-opab 3819  df-xp 4351 This theorem is referenced by:  relfld  4846
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