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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 4680 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) | |
2 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | dfsn2 4570 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | wununi.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
5 | 2, 4, 4 | wunpr 10119 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈) |
6 | 3, 5 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
7 | wunpr.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
8 | wuntp.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
9 | 2, 7, 8 | wunpr 10119 | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈) |
10 | 2, 6, 9 | wunun 10120 | . 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈) |
11 | 1, 10 | eqeltrid 2914 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cun 3931 {csn 4557 {cpr 4559 {ctp 4561 WUnicwun 10110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 df-sn 4558 df-pr 4560 df-tp 4562 df-uni 4831 df-tr 5164 df-wun 10112 |
This theorem is referenced by: catcfuccl 17357 catcxpccl 17445 |
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