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Theorem gruxp 9917
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 9916 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
2 grupw 9905 . . . 4 ((𝑈 ∈ Univ ∧ (𝐴𝐵) ∈ 𝑈) → 𝒫 (𝐴𝐵) ∈ 𝑈)
3 grupw 9905 . . . . 5 ((𝑈 ∈ Univ ∧ 𝒫 (𝐴𝐵) ∈ 𝑈) → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
4 xpsspw 5436 . . . . . 6 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
5 gruss 9906 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈 ∧ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)) → (𝐴 × 𝐵) ∈ 𝑈)
64, 5mp3an3 1575 . . . . 5 ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
73, 6syldan 586 . . . 4 ((𝑈 ∈ Univ ∧ 𝒫 (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
82, 7syldan 586 . . 3 ((𝑈 ∈ Univ ∧ (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
983ad2antl1 1237 . 2 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) ∧ (𝐴𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
101, 9mpdan 679 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108  wcel 2157  cun 3767  wss 3769  𝒫 cpw 4349   × cxp 5310  Univcgru 9900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-gru 9901
This theorem is referenced by:  grumap  9918
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