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| 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 4923 | . . 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 10749 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) | 
| 7 | 5, 6 | wununi 10746 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) | 
| 8 | 4, 7 | eqeltrrd 2842 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {cpr 4628 ∪ cuni 4907 WUnicwun 10740 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-tr 5260 df-wun 10742 | 
| This theorem is referenced by: wuntp 10751 wunsuc 10757 wunfi 10761 wunxp 10764 wuntpos 10774 wunsets 17214 catcoppccl 18162 | 
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