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| Mirrors > Home > MPE Home > Th. List > wuntr | Structured version Visualization version GIF version | ||
| Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| wuntr | ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iswun 10744 | . . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
| 2 | 1 | ibi 267 | . 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
| 3 | 2 | simp1d 1143 | 1 ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 Tr wtr 5259 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-wun 10742 |
| This theorem is referenced by: wunelss 10748 intwun 10775 |
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