Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
⊢ ∅ ∈ V |
2 |
1
|
eldm |
⊢ ( ∅ ∈ dom 𝐹 ↔ ∃ 𝑦 ∅ 𝐹 𝑦 ) |
3 |
|
brtpos0 |
⊢ ( 𝑦 ∈ V → ( ∅ tpos 𝐹 𝑦 ↔ ∅ 𝐹 𝑦 ) ) |
4 |
3
|
elv |
⊢ ( ∅ tpos 𝐹 𝑦 ↔ ∅ 𝐹 𝑦 ) |
5 |
|
0nelrel0 |
⊢ ( Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹 ) |
6 |
|
vex |
⊢ 𝑦 ∈ V |
7 |
1 6
|
breldm |
⊢ ( ∅ tpos 𝐹 𝑦 → ∅ ∈ dom tpos 𝐹 ) |
8 |
5 7
|
nsyl3 |
⊢ ( ∅ tpos 𝐹 𝑦 → ¬ Rel dom tpos 𝐹 ) |
9 |
4 8
|
sylbir |
⊢ ( ∅ 𝐹 𝑦 → ¬ Rel dom tpos 𝐹 ) |
10 |
9
|
exlimiv |
⊢ ( ∃ 𝑦 ∅ 𝐹 𝑦 → ¬ Rel dom tpos 𝐹 ) |
11 |
2 10
|
sylbi |
⊢ ( ∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹 ) |
12 |
11
|
con2i |
⊢ ( Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹 ) |
13 |
|
vex |
⊢ 𝑥 ∈ V |
14 |
13
|
eldm |
⊢ ( 𝑥 ∈ dom tpos 𝐹 ↔ ∃ 𝑦 𝑥 tpos 𝐹 𝑦 ) |
15 |
|
relcnv |
⊢ Rel ◡ dom 𝐹 |
16 |
|
df-rel |
⊢ ( Rel ◡ dom 𝐹 ↔ ◡ dom 𝐹 ⊆ ( V × V ) ) |
17 |
15 16
|
mpbi |
⊢ ◡ dom 𝐹 ⊆ ( V × V ) |
18 |
17
|
sseli |
⊢ ( 𝑥 ∈ ◡ dom 𝐹 → 𝑥 ∈ ( V × V ) ) |
19 |
18
|
a1i |
⊢ ( ( ¬ ∅ ∈ dom 𝐹 ∧ 𝑥 tpos 𝐹 𝑦 ) → ( 𝑥 ∈ ◡ dom 𝐹 → 𝑥 ∈ ( V × V ) ) ) |
20 |
|
elsni |
⊢ ( 𝑥 ∈ { ∅ } → 𝑥 = ∅ ) |
21 |
20
|
breq1d |
⊢ ( 𝑥 ∈ { ∅ } → ( 𝑥 tpos 𝐹 𝑦 ↔ ∅ tpos 𝐹 𝑦 ) ) |
22 |
1 6
|
breldm |
⊢ ( ∅ 𝐹 𝑦 → ∅ ∈ dom 𝐹 ) |
23 |
22
|
pm2.24d |
⊢ ( ∅ 𝐹 𝑦 → ( ¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ ( V × V ) ) ) |
24 |
4 23
|
sylbi |
⊢ ( ∅ tpos 𝐹 𝑦 → ( ¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ ( V × V ) ) ) |
25 |
21 24
|
syl6bi |
⊢ ( 𝑥 ∈ { ∅ } → ( 𝑥 tpos 𝐹 𝑦 → ( ¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ ( V × V ) ) ) ) |
26 |
25
|
com3l |
⊢ ( 𝑥 tpos 𝐹 𝑦 → ( ¬ ∅ ∈ dom 𝐹 → ( 𝑥 ∈ { ∅ } → 𝑥 ∈ ( V × V ) ) ) ) |
27 |
26
|
impcom |
⊢ ( ( ¬ ∅ ∈ dom 𝐹 ∧ 𝑥 tpos 𝐹 𝑦 ) → ( 𝑥 ∈ { ∅ } → 𝑥 ∈ ( V × V ) ) ) |
28 |
|
brtpos2 |
⊢ ( 𝑦 ∈ V → ( 𝑥 tpos 𝐹 𝑦 ↔ ( 𝑥 ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { 𝑥 } 𝐹 𝑦 ) ) ) |
29 |
6 28
|
ax-mp |
⊢ ( 𝑥 tpos 𝐹 𝑦 ↔ ( 𝑥 ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ∧ ∪ ◡ { 𝑥 } 𝐹 𝑦 ) ) |
30 |
29
|
simplbi |
⊢ ( 𝑥 tpos 𝐹 𝑦 → 𝑥 ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ) |
31 |
|
elun |
⊢ ( 𝑥 ∈ ( ◡ dom 𝐹 ∪ { ∅ } ) ↔ ( 𝑥 ∈ ◡ dom 𝐹 ∨ 𝑥 ∈ { ∅ } ) ) |
32 |
30 31
|
sylib |
⊢ ( 𝑥 tpos 𝐹 𝑦 → ( 𝑥 ∈ ◡ dom 𝐹 ∨ 𝑥 ∈ { ∅ } ) ) |
33 |
32
|
adantl |
⊢ ( ( ¬ ∅ ∈ dom 𝐹 ∧ 𝑥 tpos 𝐹 𝑦 ) → ( 𝑥 ∈ ◡ dom 𝐹 ∨ 𝑥 ∈ { ∅ } ) ) |
34 |
19 27 33
|
mpjaod |
⊢ ( ( ¬ ∅ ∈ dom 𝐹 ∧ 𝑥 tpos 𝐹 𝑦 ) → 𝑥 ∈ ( V × V ) ) |
35 |
34
|
ex |
⊢ ( ¬ ∅ ∈ dom 𝐹 → ( 𝑥 tpos 𝐹 𝑦 → 𝑥 ∈ ( V × V ) ) ) |
36 |
35
|
exlimdv |
⊢ ( ¬ ∅ ∈ dom 𝐹 → ( ∃ 𝑦 𝑥 tpos 𝐹 𝑦 → 𝑥 ∈ ( V × V ) ) ) |
37 |
14 36
|
syl5bi |
⊢ ( ¬ ∅ ∈ dom 𝐹 → ( 𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ ( V × V ) ) ) |
38 |
37
|
ssrdv |
⊢ ( ¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ ( V × V ) ) |
39 |
|
df-rel |
⊢ ( Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ ( V × V ) ) |
40 |
38 39
|
sylibr |
⊢ ( ¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹 ) |
41 |
12 40
|
impbii |
⊢ ( Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹 ) |