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| Mirrors > Home > MPE Home > Th. List > gruelss | Structured version Visualization version GIF version | ||
| Description: A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| gruelss | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grutr 10834 | . 2 ⊢ (𝑈 ∈ Univ → Tr 𝑈) | |
| 2 | trss 5269 | . . 3 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈)) | |
| 3 | 2 | imp 406 | . 2 ⊢ ((Tr 𝑈 ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | 
| 4 | 1, 3 | sylan 580 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ⊆ wss 3950 Tr wtr 5258 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 | 
| 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-ss 3967 df-nul 4333 df-if 4525 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: gruss 10837 gruuni 10841 gruel 10844 grur1a 10860 grur1 10861 | 
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