Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑤 ∈ V |
2 |
1
|
elpr |
⊢ ( 𝑤 ∈ { 𝑥 , 𝑦 } ↔ ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) ) |
3 |
2
|
biimpri |
⊢ ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) |
4 |
3
|
rgenw |
⊢ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) |
5 |
|
eleq2 |
⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( 𝑤 ∈ 𝑧 ↔ 𝑤 ∈ { 𝑥 , 𝑦 } ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ↔ ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) ) |
7 |
6
|
ralbidv |
⊢ ( 𝑧 = { 𝑥 , 𝑦 } → ( ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) ) |
8 |
7
|
rspcev |
⊢ ( ( { 𝑥 , 𝑦 } ∈ 𝑀 ∧ ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ { 𝑥 , 𝑦 } ) ) → ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) |
9 |
4 8
|
mpan2 |
⊢ ( { 𝑥 , 𝑦 } ∈ 𝑀 → ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) |
10 |
9
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 { 𝑥 , 𝑦 } ∈ 𝑀 → ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ∃ 𝑧 ∈ 𝑀 ∀ 𝑤 ∈ 𝑀 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) |