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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 10115 | . . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
2 | 1 | ibi 270 | . 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
3 | 2 | simp1d 1139 | 1 ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∅c0 4243 𝒫 cpw 4497 {cpr 4527 ∪ cuni 4800 Tr wtr 5136 WUnicwun 10111 |
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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-tr 5137 df-wun 10113 |
This theorem is referenced by: wunelss 10119 intwun 10146 |
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