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Mirrors > Home > MPE Home > Th. List > wunint | Structured version Visualization version GIF version |
Description: A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunint | ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑈 ∈ WUni) |
3 | wununi.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 3 | wununi 10117 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
5 | 4 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈) |
6 | intssuni 4860 | . . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
7 | 6 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
8 | 2, 5, 7 | wunss 10123 | 1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∩ cint 4838 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-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-uni 4801 df-int 4839 df-tr 5137 df-wun 10113 |
This theorem is referenced by: (None) |
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