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| Description: A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| gruuni | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uniiun 5057 | . 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 2 | gruelss 10835 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | |
| 3 | dfss3 3971 | . . . 4 ⊢ (𝐴 ⊆ 𝑈 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝑈) | |
| 4 | 2, 3 | sylib 218 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝑈) | 
| 5 | gruiun 10840 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝑈) | |
| 6 | 4, 5 | mpd3an3 1463 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝑈) | 
| 7 | 1, 6 | eqeltrid 2844 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 ∪ cuni 4906 ∪ ciun 4990 Univcgru 10831 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-gru 10832 | 
| This theorem is referenced by: gruwun 10854 gruina 10859 grumnudlem 44309 | 
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