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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnusnd | Structured version Visualization version GIF version |
Description: Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnusnd.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))))} |
mnusnd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnusnd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
mnusnd | ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnusnd.1 | . 2 ⊢ 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 | mnusnd.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | mnusnd.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 2, 3 | mnupwd 40126 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
5 | snsspw 4686 | . . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → {𝐴} ⊆ 𝒫 𝐴) |
7 | 1, 2, 4, 6 | mnussd 40122 | 1 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1520 = wceq 1522 ∈ wcel 2081 {cab 2775 ∀wral 3105 ∃wrex 3106 ⊆ wss 3863 𝒫 cpw 4457 {csn 4476 ∪ cuni 4749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-v 3439 df-dif 3866 df-in 3870 df-ss 3878 df-nul 4216 df-pw 4459 df-sn 4477 df-uni 4750 |
This theorem is referenced by: mnuprdlem4 40134 |
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