Metamath Proof Explorer
Description: A minimal universe contains pairs of subsets of an element of the
universe. (Contributed by Rohan Ridenour, 13-Aug-2023)
|
|
Ref |
Expression |
|
Hypotheses |
mnuprssd.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 ) ) ) ) } |
|
|
mnuprssd.2 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
|
|
mnuprssd.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑈 ) |
|
|
mnuprssd.4 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
|
|
mnuprssd.5 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
|
Assertion |
mnuprssd |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mnuprssd.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 ) ) ) ) } |
2 |
|
mnuprssd.2 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
3 |
|
mnuprssd.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑈 ) |
4 |
|
mnuprssd.4 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
5 |
|
mnuprssd.5 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
6 |
1 2 3
|
mnupwd |
⊢ ( 𝜑 → 𝒫 𝐶 ∈ 𝑈 ) |
7 |
3 4
|
sselpwd |
⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝐶 ) |
8 |
3 5
|
sselpwd |
⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐶 ) |
9 |
7 8
|
prssd |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ) |
10 |
1 2 6 9
|
mnussd |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 ) |