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Mirrors > Home > MPE Home > Th. List > gruxp | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruxp | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gruun 10807 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈) | |
2 | grupw 10796 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ (𝐴 ∪ 𝐵) ∈ 𝑈) → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) | |
3 | grupw 10796 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) | |
4 | xpsspw 5809 | . . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
5 | gruss 10797 | . . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈 ∧ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)) → (𝐴 × 𝐵) ∈ 𝑈) | |
6 | 4, 5 | mp3an3 1449 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
7 | 3, 6 | syldan 590 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
8 | 2, 7 | syldan 590 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ (𝐴 ∪ 𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
9 | 8 | 3ad2antl1 1184 | . 2 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) ∧ (𝐴 ∪ 𝐵) ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
10 | 1, 9 | mpdan 684 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2105 ∪ cun 3946 ⊆ wss 3948 𝒫 cpw 4602 × cxp 5674 Univcgru 10791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-gru 10792 |
This theorem is referenced by: grumap 10809 |
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