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| Description: Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| gruss | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elpw2g 5332 | . . . 4 ⊢ (𝐴 ∈ 𝑈 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
| 2 | 1 | adantl 481 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) | 
| 3 | grupw 10836 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) | |
| 4 | gruelss 10835 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 𝐴 ∈ 𝑈) → 𝒫 𝐴 ⊆ 𝑈) | |
| 5 | 3, 4 | syldan 591 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ⊆ 𝑈) | 
| 6 | 5 | sseld 3981 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐵 ∈ 𝒫 𝐴 → 𝐵 ∈ 𝑈)) | 
| 7 | 2, 6 | sylbird 260 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐵 ⊆ 𝐴 → 𝐵 ∈ 𝑈)) | 
| 8 | 7 | 3impia 1117 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 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-ext 2707 ax-sep 5295 | 
| 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 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-br 5143 df-tr 5259 df-iota 6513 df-fv 6568 df-ov 7435 df-gru 10832 | 
| This theorem is referenced by: grurn 10842 gruima 10843 gruxp 10848 grumap 10849 gruixp 10850 gruiin 10851 grudomon 10858 gruina 10859 gru0eld 44253 grur1cld 44256 grurankrcld 44258 grumnudlem 44309 | 
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