Step |
Hyp |
Ref |
Expression |
1 |
|
sadval.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ0 ) |
2 |
|
sadval.b |
⊢ ( 𝜑 → 𝐵 ⊆ ℕ0 ) |
3 |
|
sadval.c |
⊢ 𝐶 = seq 0 ( ( 𝑐 ∈ 2o , 𝑚 ∈ ℕ0 ↦ if ( cadd ( 𝑚 ∈ 𝐴 , 𝑚 ∈ 𝐵 , ∅ ∈ 𝑐 ) , 1o , ∅ ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) |
4 |
|
noel |
⊢ ¬ ∅ ∈ ∅ |
5 |
3
|
fveq1i |
⊢ ( 𝐶 ‘ 0 ) = ( seq 0 ( ( 𝑐 ∈ 2o , 𝑚 ∈ ℕ0 ↦ if ( cadd ( 𝑚 ∈ 𝐴 , 𝑚 ∈ 𝐵 , ∅ ∈ 𝑐 ) , 1o , ∅ ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) |
6 |
|
0z |
⊢ 0 ∈ ℤ |
7 |
|
seq1 |
⊢ ( 0 ∈ ℤ → ( seq 0 ( ( 𝑐 ∈ 2o , 𝑚 ∈ ℕ0 ↦ if ( cadd ( 𝑚 ∈ 𝐴 , 𝑚 ∈ 𝐵 , ∅ ∈ 𝑐 ) , 1o , ∅ ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) = ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ‘ 0 ) ) |
8 |
6 7
|
ax-mp |
⊢ ( seq 0 ( ( 𝑐 ∈ 2o , 𝑚 ∈ ℕ0 ↦ if ( cadd ( 𝑚 ∈ 𝐴 , 𝑚 ∈ 𝐵 , ∅ ∈ 𝑐 ) , 1o , ∅ ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) = ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ‘ 0 ) |
9 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
10 |
|
iftrue |
⊢ ( 𝑛 = 0 → if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) = ∅ ) |
11 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) |
12 |
|
0ex |
⊢ ∅ ∈ V |
13 |
10 11 12
|
fvmpt |
⊢ ( 0 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ‘ 0 ) = ∅ ) |
14 |
9 13
|
ax-mp |
⊢ ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ‘ 0 ) = ∅ |
15 |
5 8 14
|
3eqtri |
⊢ ( 𝐶 ‘ 0 ) = ∅ |
16 |
15
|
eleq2i |
⊢ ( ∅ ∈ ( 𝐶 ‘ 0 ) ↔ ∅ ∈ ∅ ) |
17 |
4 16
|
mtbir |
⊢ ¬ ∅ ∈ ( 𝐶 ‘ 0 ) |
18 |
17
|
a1i |
⊢ ( 𝜑 → ¬ ∅ ∈ ( 𝐶 ‘ 0 ) ) |