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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mnussd | Structured version Visualization version GIF version | ||
| Description: Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| mnussd.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))))} | 
| mnussd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) | 
| mnussd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) | 
| mnussd.4 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | 
| Ref | Expression | 
|---|---|
| mnussd | ⊢ (𝜑 → 𝐵 ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mnussd.1 | . . . 4 ⊢ 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))))} | |
| 2 | mnussd.2 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
| 3 | mnussd.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 4 | 1, 2, 3 | mnuop123d 44286 | . . 3 ⊢ (𝜑 → (𝒫 𝐴 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))) | 
| 5 | 4 | simpld 494 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝑈) | 
| 6 | mnussd.4 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 7 | 3, 6 | sselpwd 5327 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) | 
| 8 | 5, 7 | sseldd 3983 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 df-pw 4601 df-uni 4907 | 
| This theorem is referenced by: mnuss2d 44288 mnu0eld 44289 mnusnd 44292 mnuprssd 44293 mnuprdlem4 44299 mnutrcld 44303 | 
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