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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 4411 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | wununi.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 2, 3, 3 | wunpr 9868 | . 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
5 | 1, 4 | syl5eqel 2863 | 1 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {csn 4398 {cpr 4400 WUnicwun 9859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-v 3400 df-un 3797 df-in 3799 df-ss 3806 df-sn 4399 df-pr 4401 df-uni 4674 df-tr 4990 df-wun 9861 |
This theorem is referenced by: wunsuc 9876 wunfi 9880 wunop 9881 wuntpos 9893 wunsets 16307 1strwunbndx 16384 catcoppccl 17154 |
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