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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 10011 | . . 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 259 | . 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 487 | 1 ⊢ (𝑈 ∈ Univ → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 ∀wral 3083 𝒫 cpw 4417 {cpr 4438 ∪ cuni 4709 Tr wtr 5027 ran crn 5405 (class class class)co 6975 ↑𝑚 cmap 8205 Univcgru 10009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-tr 5028 df-iota 6150 df-fv 6194 df-ov 6978 df-gru 10010 |
This theorem is referenced by: gruelss 10013 gruwun 10032 intgru 10033 gruina 10037 grur1 10039 grutsk 10041 |
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