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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 10407 | . 2 ⊢ (𝑈 ∈ Univ → Tr 𝑈) | |
2 | trss 5170 | . . 3 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈)) | |
3 | 2 | imp 410 | . 2 ⊢ ((Tr 𝑈 ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) |
4 | 1, 3 | sylan 583 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ⊆ wss 3866 Tr wtr 5161 Univcgru 10404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-tr 5162 df-iota 6338 df-fv 6388 df-ov 7216 df-gru 10405 |
This theorem is referenced by: gruss 10410 gruuni 10414 gruel 10417 grur1a 10433 grur1 10434 |
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