|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > grutr | Structured version Visualization version GIF version | ||
| Description: A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| grutr | ⊢ (𝑈 ∈ Univ → Tr 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elgrug 10833 | . . 3 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))) | |
| 2 | 1 | ibi 267 | . 2 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))) | 
| 3 | 2 | simpld 494 | 1 ⊢ (𝑈 ∈ Univ → Tr 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 ∀wral 3060 𝒫 cpw 4599 {cpr 4627 ∪ cuni 4906 Tr wtr 5258 ran crn 5685 (class class class)co 7432 ↑m cmap 8867 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: gruelss 10835 gruwun 10854 intgru 10855 gruina 10859 grur1 10861 grutsk 10863 | 
| Copyright terms: Public domain | W3C validator |