Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 <Q 𝑦 ↔ 𝑤 <Q 𝑦 ) ) |
2 |
1
|
anbi1d |
⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) ↔ ( 𝑤 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
3 |
2
|
exbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) ↔ ∃ 𝑦 ( 𝑤 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
4 |
3
|
cbvabv |
⊢ { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } = { 𝑤 ∣ ∃ 𝑦 ( 𝑤 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } |
5 |
4
|
reclem2pr |
⊢ ( 𝐴 ∈ P → { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ∈ P ) |
6 |
4
|
reclem4pr |
⊢ ( 𝐴 ∈ P → ( 𝐴 ·P { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ) = 1P ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } → ( 𝐴 ·P 𝑥 ) = ( 𝐴 ·P { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑥 = { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } → ( ( 𝐴 ·P 𝑥 ) = 1P ↔ ( 𝐴 ·P { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ) = 1P ) ) |
9 |
8
|
rspcev |
⊢ ( ( { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ∈ P ∧ ( 𝐴 ·P { 𝑧 ∣ ∃ 𝑦 ( 𝑧 <Q 𝑦 ∧ ¬ ( *Q ‘ 𝑦 ) ∈ 𝐴 ) } ) = 1P ) → ∃ 𝑥 ∈ P ( 𝐴 ·P 𝑥 ) = 1P ) |
10 |
5 6 9
|
syl2anc |
⊢ ( 𝐴 ∈ P → ∃ 𝑥 ∈ P ( 𝐴 ·P 𝑥 ) = 1P ) |