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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 10117 | . . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
4 | 1, 3 | wunelss 10118 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝑈) |
5 | wunss.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | 2, 5 | sselpwd 5221 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
7 | 4, 6 | sseldd 3965 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 WUnicwun 10110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 df-uni 4831 df-tr 5164 df-wun 10112 |
This theorem is referenced by: wunin 10123 wundif 10124 wunint 10125 wun0 10128 wunom 10130 wunxp 10134 wunpm 10135 wunmap 10136 wundm 10138 wunrn 10139 wuncnv 10140 wunres 10141 wunfv 10142 wunco 10143 wuntpos 10144 wuncn 10580 wunndx 16492 wunstr 16495 wunfunc 17157 |
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