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Mirrors > Home > MPE Home > Th. List > wundif | Structured version Visualization version GIF version |
Description: A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wundif | ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | difssd 4023 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | |
4 | 1, 2, 3 | wunss 10212 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∖ cdif 3840 WUnicwun 10200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2710 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rab 3062 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-pw 4490 df-uni 4797 df-tr 5137 df-wun 10202 |
This theorem is referenced by: wuncn 10670 |
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