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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 10698 | . . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
2 | 1 | ibi 266 | . 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
3 | 2 | simp1d 1142 | 1 ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∅c0 4322 𝒫 cpw 4602 {cpr 4630 ∪ cuni 4908 Tr wtr 5265 WUnicwun 10694 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-tr 5266 df-wun 10696 |
This theorem is referenced by: wunelss 10702 intwun 10729 |
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