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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 41839 | . . 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 495 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝑈) |
6 | mnussd.4 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
7 | 3, 6 | sselpwd 5249 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
8 | 5, 7 | sseldd 3922 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1537 = wceq 1539 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 𝒫 cpw 4534 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5222 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-in 3894 df-ss 3904 df-pw 4536 df-uni 4841 |
This theorem is referenced by: mnuss2d 41841 mnu0eld 41842 mnusnd 41845 mnuprssd 41846 mnuprdlem4 41852 mnutrcld 41856 |
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