Step |
Hyp |
Ref |
Expression |
1 |
|
dfint2 |
⊢ ∩ 𝐴 = { 𝑥 ∣ ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 } |
2 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 ◡ ( V ∖ E ) 𝑥 ↔ ¬ ∃ 𝑦 ∈ 𝐴 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
3 |
|
vex |
⊢ 𝑦 ∈ V |
4 |
|
vex |
⊢ 𝑥 ∈ V |
5 |
3 4
|
brcnv |
⊢ ( 𝑦 ◡ ( V ∖ E ) 𝑥 ↔ 𝑥 ( V ∖ E ) 𝑦 ) |
6 |
|
brv |
⊢ 𝑥 V 𝑦 |
7 |
|
brdif |
⊢ ( 𝑥 ( V ∖ E ) 𝑦 ↔ ( 𝑥 V 𝑦 ∧ ¬ 𝑥 E 𝑦 ) ) |
8 |
6 7
|
mpbiran |
⊢ ( 𝑥 ( V ∖ E ) 𝑦 ↔ ¬ 𝑥 E 𝑦 ) |
9 |
5 8
|
bitr2i |
⊢ ( ¬ 𝑥 E 𝑦 ↔ 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
10 |
9
|
con1bii |
⊢ ( ¬ 𝑦 ◡ ( V ∖ E ) 𝑥 ↔ 𝑥 E 𝑦 ) |
11 |
|
epel |
⊢ ( 𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦 ) |
12 |
10 11
|
bitr2i |
⊢ ( 𝑥 ∈ 𝑦 ↔ ¬ 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
13 |
12
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
14 |
|
eldif |
⊢ ( 𝑥 ∈ ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) ↔ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ( ◡ ( V ∖ E ) “ 𝐴 ) ) ) |
15 |
4 14
|
mpbiran |
⊢ ( 𝑥 ∈ ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) ↔ ¬ 𝑥 ∈ ( ◡ ( V ∖ E ) “ 𝐴 ) ) |
16 |
4
|
elima |
⊢ ( 𝑥 ∈ ( ◡ ( V ∖ E ) “ 𝐴 ) ↔ ∃ 𝑦 ∈ 𝐴 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
17 |
15 16
|
xchbinx |
⊢ ( 𝑥 ∈ ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) ↔ ¬ ∃ 𝑦 ∈ 𝐴 𝑦 ◡ ( V ∖ E ) 𝑥 ) |
18 |
2 13 17
|
3bitr4ri |
⊢ ( 𝑥 ∈ ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) ↔ ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ) |
19 |
18
|
abbi2i |
⊢ ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) = { 𝑥 ∣ ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 } |
20 |
1 19
|
eqtr4i |
⊢ ∩ 𝐴 = ( V ∖ ( ◡ ( V ∖ E ) “ 𝐴 ) ) |