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Mirrors > Home > MPE Home > Th. List > wunsn | Structured version Visualization version GIF version |
Description: A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunsn | ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4580 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | wununi.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 2, 3, 3 | wunpr 10131 | . 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
5 | 1, 4 | eqeltrid 2917 | 1 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {csn 4567 {cpr 4569 WUnicwun 10122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-v 3496 df-un 3941 df-in 3943 df-ss 3952 df-sn 4568 df-pr 4570 df-uni 4839 df-tr 5173 df-wun 10124 |
This theorem is referenced by: wunsuc 10139 wunfi 10143 wunop 10144 wuntpos 10156 wunsets 16524 1strwunbndx 16600 catcoppccl 17368 |
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