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Mirrors > Home > MPE Home > Th. List > wunin | Structured version Visualization version GIF version |
Description: A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunin | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | inss1 4155 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴) |
5 | 1, 2, 4 | wunss 10123 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 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-11 2158 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 df-uni 4801 df-tr 5137 df-wun 10113 |
This theorem is referenced by: wunress 16556 |
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