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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuund | Structured version Visualization version GIF version | ||
| Description: Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| mnuund.1 | ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} | 
| mnuund.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) | 
| mnuund.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) | 
| mnuund.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) | 
| Ref | Expression | 
|---|---|
| mnuund | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mnuund.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 2 | mnuund.4 | . . 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 | mnuund.1 | . . 3 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} | |
| 6 | mnuund.2 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
| 7 | 5, 6, 1, 2 | mnuprd 44295 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) | 
| 8 | 5, 6, 7 | mnuunid 44296 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) | 
| 9 | 4, 8 | eqeltrrd 2842 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 | 
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 | 
| This theorem is referenced by: (None) | 
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