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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 10790 | . 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 579 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3943 Tr wtr 5258 Univcgru 10787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-tr 5259 df-iota 6489 df-fv 6545 df-ov 7408 df-gru 10788 |
This theorem is referenced by: gruss 10793 gruuni 10797 gruel 10800 grur1a 10816 grur1 10817 |
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