![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wunss | Structured version Visualization version GIF version |
Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunss.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
wunss | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wununi.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wunpw 10745 | . . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
4 | 1, 3 | wunelss 10746 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝑈) |
5 | wunss.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | 2, 5 | sselpwd 5334 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
7 | 4, 6 | sseldd 3996 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 WUnicwun 10738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 df-uni 4913 df-tr 5266 df-wun 10740 |
This theorem is referenced by: wunin 10751 wundif 10752 wunint 10753 wun0 10756 wunom 10758 wunxp 10762 wunpm 10763 wunmap 10764 wundm 10766 wunrn 10767 wuncnv 10768 wunres 10769 wunfv 10770 wunco 10771 wuntpos 10772 wuncn 11208 wunstr 17222 wunndx 17229 wunfunc 17952 wunfuncOLD 17953 |
Copyright terms: Public domain | W3C validator |