![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wunelss | Structured version Visualization version GIF version |
Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunelss | ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wuntr 10116 | . . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → Tr 𝑈) |
4 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
5 | trss 5145 | . 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈)) | |
6 | 3, 4, 5 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 Tr wtr 5136 WUnicwun 10111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-tr 5137 df-wun 10113 |
This theorem is referenced by: wunss 10123 wunf 10138 wuncval2 10158 |
Copyright terms: Public domain | W3C validator |