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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 481 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑈 ∈ WUni) |
3 | wununi.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 3 | wununi 10563 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
5 | 4 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈) |
6 | intssuni 4918 | . . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
7 | 6 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴) |
8 | 2, 5, 7 | wunss 10569 | 1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 2940 ⊆ wss 3898 ∅c0 4269 ∪ cuni 4852 ∩ cint 4894 WUnicwun 10557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 ax-sep 5243 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-in 3905 df-ss 3915 df-nul 4270 df-pw 4549 df-uni 4853 df-int 4895 df-tr 5210 df-wun 10559 |
This theorem is referenced by: (None) |
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