Step |
Hyp |
Ref |
Expression |
1 |
|
eldifsn |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ↔ ( 𝑃 ∈ 𝒫 𝑉 ∧ 𝑃 ≠ ∅ ) ) |
2 |
|
hashle2pr |
⊢ ( ( 𝑃 ∈ 𝒫 𝑉 ∧ 𝑃 ≠ ∅ ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ) ) |
3 |
1 2
|
sylbi |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ) ) |
4 |
|
eldifi |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → 𝑃 ∈ 𝒫 𝑉 ) |
5 |
|
eleq1 |
⊢ ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑃 ∈ 𝒫 𝑉 ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) ) |
6 |
|
prelpw |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ↔ { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 ) ) |
7 |
6
|
biimprd |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ) |
8 |
7
|
el2v |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) |
9 |
5 8
|
syl6bi |
⊢ ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑃 ∈ 𝒫 𝑉 → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ) |
10 |
4 9
|
syl5com |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ) |
11 |
10
|
pm4.71rd |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑃 = { 𝑎 , 𝑏 } ↔ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) ) |
12 |
11
|
2exbidv |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎 ∃ 𝑏 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) ) |
13 |
|
r2ex |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ) |
14 |
13
|
bicomi |
⊢ ( ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) |
15 |
14
|
a1i |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ∃ 𝑎 ∃ 𝑏 ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑃 = { 𝑎 , 𝑏 } ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) ) |
16 |
3 12 15
|
3bitrd |
⊢ ( 𝑃 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑃 ) ≤ 2 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 𝑃 = { 𝑎 , 𝑏 } ) ) |