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Mirrors > Home > MPE Home > Th. List > gruss | Structured version Visualization version GIF version |
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 5339 | . . . 4 ⊢ (𝐴 ∈ 𝑈 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) |
3 | grupw 10833 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) | |
4 | gruelss 10832 | . . . . 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 3994 | . . 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 1116 | 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 2106 ⊆ wss 3963 𝒫 cpw 4605 Univcgru 10828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-tr 5266 df-iota 6516 df-fv 6571 df-ov 7434 df-gru 10829 |
This theorem is referenced by: grurn 10839 gruima 10840 gruxp 10845 grumap 10846 gruixp 10847 gruiin 10848 grudomon 10855 gruina 10856 gru0eld 44225 grur1cld 44228 grurankrcld 44230 grumnudlem 44281 |
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