Step |
Hyp |
Ref |
Expression |
1 |
|
fbasne0 |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝐵 ) → 𝐹 ≠ ∅ ) |
2 |
|
n0 |
⊢ ( 𝐹 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐹 ) |
3 |
1 2
|
sylib |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝐵 ) → ∃ 𝑥 𝑥 ∈ 𝐹 ) |
4 |
|
ssv |
⊢ 𝑥 ⊆ V |
5 |
4
|
jctr |
⊢ ( 𝑥 ∈ 𝐹 → ( 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V ) ) |
6 |
5
|
eximi |
⊢ ( ∃ 𝑥 𝑥 ∈ 𝐹 → ∃ 𝑥 ( 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V ) ) |
7 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ V ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V ) ) |
8 |
6 7
|
sylibr |
⊢ ( ∃ 𝑥 𝑥 ∈ 𝐹 → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ V ) |
9 |
3 8
|
syl |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝐵 ) → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ V ) |
10 |
|
inteq |
⊢ ( 𝐴 = ∅ → ∩ 𝐴 = ∩ ∅ ) |
11 |
|
int0 |
⊢ ∩ ∅ = V |
12 |
10 11
|
eqtrdi |
⊢ ( 𝐴 = ∅ → ∩ 𝐴 = V ) |
13 |
12
|
sseq2d |
⊢ ( 𝐴 = ∅ → ( 𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ⊆ V ) ) |
14 |
13
|
rexbidv |
⊢ ( 𝐴 = ∅ → ( ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ↔ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ V ) ) |
15 |
9 14
|
syl5ibrcom |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝐵 ) → ( 𝐴 = ∅ → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) → ( 𝐴 = ∅ → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) ) |
17 |
|
simpl1 |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐹 ∈ ( fBas ‘ 𝐵 ) ) |
18 |
|
simpl2 |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ 𝐹 ) |
19 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ≠ ∅ ) |
20 |
|
simpl3 |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ∈ Fin ) |
21 |
|
elfir |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ ( 𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ 𝐴 ∈ ( fi ‘ 𝐹 ) ) |
22 |
17 18 19 20 21
|
syl13anc |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → ∩ 𝐴 ∈ ( fi ‘ 𝐹 ) ) |
23 |
|
fbssfi |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ ∩ 𝐴 ∈ ( fi ‘ 𝐹 ) ) → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) |
24 |
17 22 23
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) |
25 |
24
|
ex |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) → ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) ) |
26 |
16 25
|
pm2.61dne |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin ) → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ) |