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Mirrors > Home > MPE Home > Th. List > wuntp | Structured version Visualization version GIF version |
Description: A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunpr.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
wuntp.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
wuntp | ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpass 4717 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | dfsn2 4603 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | wununi.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
5 | 2, 4, 4 | wunpr 10653 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
6 | 3, 5 | eqeltrid 2838 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | wunpr.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
8 | wuntp.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
9 | 2, 7, 8 | wunpr 10653 | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈) |
10 | 2, 6, 9 | wunun 10654 | . 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈) |
11 | 1, 10 | eqeltrid 2838 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∪ cun 3912 {csn 4590 {cpr 4592 {ctp 4594 WUnicwun 10644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-v 3449 df-un 3919 df-in 3921 df-ss 3931 df-sn 4591 df-pr 4593 df-tp 4595 df-uni 4870 df-tr 5227 df-wun 10646 |
This theorem is referenced by: catcfuccl 18013 catcfucclOLD 18014 catcxpccl 18103 catcxpcclOLD 18104 |
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