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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 10823 | . . 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 266 | . 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 493 | 1 ⊢ (𝑈 ∈ Univ → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 ∀wral 3051 𝒫 cpw 4597 {cpr 4625 ∪ cuni 4905 Tr wtr 5260 ran crn 5673 (class class class)co 7413 ↑m cmap 8844 Univcgru 10821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-tr 5261 df-iota 6495 df-fv 6551 df-ov 7416 df-gru 10822 |
This theorem is referenced by: gruelss 10825 gruwun 10844 intgru 10845 gruina 10849 grur1 10851 grutsk 10853 |
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