![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4538 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | wununi.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 2, 3, 3 | wunpr 10120 | . 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
5 | 1, 4 | eqeltrid 2894 | 1 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 {csn 4525 {cpr 4527 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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-tr 5137 df-wun 10113 |
This theorem is referenced by: wunsuc 10128 wunfi 10132 wunop 10133 wuntpos 10145 wunsets 16516 1strwunbndx 16592 catcoppccl 17360 |
Copyright terms: Public domain | W3C validator |