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Theorem gruxp 9614
Description: A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruxp ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)

Proof of Theorem gruxp
StepHypRef Expression
1 gruun 9613 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
2 grupw 9602 . . . 4 ((𝑈 ∈ Univ ∧ (𝐴𝐵) ∈ 𝑈) → 𝒫 (𝐴𝐵) ∈ 𝑈)
3 grupw 9602 . . . . 5 ((𝑈 ∈ Univ ∧ 𝒫 (𝐴𝐵) ∈ 𝑈) → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
4 xpsspw 5223 . . . . . 6 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
5 gruss 9603 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈 ∧ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)) → (𝐴 × 𝐵) ∈ 𝑈)
64, 5mp3an3 1411 . . . . 5 ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
73, 6syldan 487 . . . 4 ((𝑈 ∈ Univ ∧ 𝒫 (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
82, 7syldan 487 . . 3 ((𝑈 ∈ Univ ∧ (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
983ad2antl1 1221 . 2 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) ∧ (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
101, 9mpdan 701 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1988  cun 3565  wss 3567  𝒫 cpw 4149   × cxp 5102  Univcgru 9597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-gru 9598
This theorem is referenced by:  grumap  9615
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