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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 4648 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | dfsn2 4538 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | wununi.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
5 | 2, 4, 4 | wunpr 10120 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
6 | 3, 5 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | wunpr.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
8 | wuntp.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
9 | 2, 7, 8 | wunpr 10120 | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈) |
10 | 2, 6, 9 | wunun 10121 | . 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈) |
11 | 1, 10 | eqeltrid 2894 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cun 3879 {csn 4525 {cpr 4527 {ctp 4529 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-3or 1085 df-3an 1086 df-tru 1541 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-tp 4530 df-uni 4801 df-tr 5137 df-wun 10113 |
This theorem is referenced by: catcfuccl 17361 catcxpccl 17449 |
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