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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 4673 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 581 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
5 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1, 2 | wunpr 9847 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
7 | 5, 6 | wununi 9844 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) |
8 | 4, 7 | eqeltrrd 2908 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∪ cun 3797 {cpr 4400 ∪ cuni 4659 WUnicwun 9838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-v 3417 df-un 3804 df-in 3806 df-ss 3813 df-sn 4399 df-pr 4401 df-uni 4660 df-tr 4977 df-wun 9840 |
This theorem is referenced by: wuntp 9849 wunsuc 9855 wunfi 9859 wunxp 9862 wuntpos 9872 wunsets 16264 catcoppccl 17111 |
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