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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 10203 | . . 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 270 | . 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 498 | 1 ⊢ (𝑈 ∈ Univ → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∀wral 3106 𝒫 cpw 4497 {cpr 4527 ∪ cuni 4800 Tr wtr 5136 ran crn 5520 (class class class)co 7135 ↑m cmap 8389 Univcgru 10201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-tr 5137 df-iota 6283 df-fv 6332 df-ov 7138 df-gru 10202 |
This theorem is referenced by: gruelss 10205 gruwun 10224 intgru 10225 gruina 10229 grur1 10231 grutsk 10233 |
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