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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 4230 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴) |
5 | 1, 2, 4 | wunss 10755 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∩ cin 3946 ⊆ wss 3947 WUnicwun 10743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-in 3954 df-ss 3964 df-pw 4609 df-uni 4914 df-tr 5271 df-wun 10745 |
This theorem is referenced by: wunress 17264 wunressOLD 17265 |
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