![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
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 43510 | . . 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 5316 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
8 | 5, 7 | sseldd 3975 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2701 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 𝒫 cpw 4594 ∪ cuni 4899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-in 3947 df-ss 3957 df-pw 4596 df-uni 4900 |
This theorem is referenced by: mnuss2d 43512 mnu0eld 43513 mnusnd 43516 mnuprssd 43517 mnuprdlem4 43523 mnutrcld 43527 |
Copyright terms: Public domain | W3C validator |