Step |
Hyp |
Ref |
Expression |
1 |
|
elin |
⊢ ( 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ↔ ( 𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin ) ) |
2 |
|
elpwi |
⊢ ( 𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹 ) |
3 |
2
|
anim1i |
⊢ ( ( 𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin ) → ( 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin ) ) |
4 |
1 3
|
sylbi |
⊢ ( 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) → ( 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin ) ) |
5 |
|
fbssint |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin ) → ∃ 𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦 ) |
6 |
5
|
3expb |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ ( 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin ) ) → ∃ 𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦 ) |
7 |
4 6
|
sylan2 |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) → ∃ 𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦 ) |
8 |
|
0nelfb |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ¬ ∅ ∈ 𝐹 ) |
9 |
8
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ 𝑧 ∈ 𝐹 ) → ¬ ∅ ∈ 𝐹 ) |
10 |
|
eleq1 |
⊢ ( 𝑧 = ∅ → ( 𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹 ) ) |
11 |
10
|
biimpcd |
⊢ ( 𝑧 ∈ 𝐹 → ( 𝑧 = ∅ → ∅ ∈ 𝐹 ) ) |
12 |
11
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ 𝑧 ∈ 𝐹 ) → ( 𝑧 = ∅ → ∅ ∈ 𝐹 ) ) |
13 |
9 12
|
mtod |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ 𝑧 ∈ 𝐹 ) → ¬ 𝑧 = ∅ ) |
14 |
|
ss0 |
⊢ ( 𝑧 ⊆ ∅ → 𝑧 = ∅ ) |
15 |
13 14
|
nsyl |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ 𝑧 ∈ 𝐹 ) → ¬ 𝑧 ⊆ ∅ ) |
16 |
15
|
adantrr |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ ( 𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦 ) ) → ¬ 𝑧 ⊆ ∅ ) |
17 |
|
sseq2 |
⊢ ( ∅ = ∩ 𝑦 → ( 𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦 ) ) |
18 |
17
|
biimprcd |
⊢ ( 𝑧 ⊆ ∩ 𝑦 → ( ∅ = ∩ 𝑦 → 𝑧 ⊆ ∅ ) ) |
19 |
18
|
ad2antll |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ ( 𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦 ) ) → ( ∅ = ∩ 𝑦 → 𝑧 ⊆ ∅ ) ) |
20 |
16 19
|
mtod |
⊢ ( ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) ∧ ( 𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦 ) ) → ¬ ∅ = ∩ 𝑦 ) |
21 |
7 20
|
rexlimddv |
⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ) → ¬ ∅ = ∩ 𝑦 ) |
22 |
21
|
nrexdv |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ¬ ∃ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ∅ = ∩ 𝑦 ) |
23 |
|
0ex |
⊢ ∅ ∈ V |
24 |
|
elfi |
⊢ ( ( ∅ ∈ V ∧ 𝐹 ∈ ( fBas ‘ 𝑋 ) ) → ( ∅ ∈ ( fi ‘ 𝐹 ) ↔ ∃ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ∅ = ∩ 𝑦 ) ) |
25 |
23 24
|
mpan |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( ∅ ∈ ( fi ‘ 𝐹 ) ↔ ∃ 𝑦 ∈ ( 𝒫 𝐹 ∩ Fin ) ∅ = ∩ 𝑦 ) ) |
26 |
22 25
|
mtbird |
⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ¬ ∅ ∈ ( fi ‘ 𝐹 ) ) |