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Mirrors > Home > MPE Home > Th. List > wunun | Structured version Visualization version GIF version |
Description: A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunpr.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
wunun | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | wunpr.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | uniprg 4925 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
5 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1, 2 | wunpr 10703 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
7 | 5, 6 | wununi 10700 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) |
8 | 4, 7 | eqeltrrd 2834 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3946 {cpr 4630 ∪ cuni 4908 WUnicwun 10694 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 df-uni 4909 df-tr 5266 df-wun 10696 |
This theorem is referenced by: wuntp 10705 wunsuc 10711 wunfi 10715 wunxp 10718 wuntpos 10728 wunsets 17109 catcoppccl 18066 catcoppcclOLD 18067 |
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