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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 10695 | . . 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 1139 | 1 ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∅c0 4314 𝒫 cpw 4594 {cpr 4622 ∪ cuni 4899 Tr wtr 5255 WUnicwun 10691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-v 3468 df-in 3947 df-ss 3957 df-uni 4900 df-tr 5256 df-wun 10693 |
This theorem is referenced by: wunelss 10699 intwun 10726 |
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