Step |
Hyp |
Ref |
Expression |
1 |
|
ordtconn.x |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ordtconn.l |
⊢ ≤ = ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) |
3 |
|
ordtconn.j |
⊢ 𝐽 = ( ordTop ‘ ≤ ) |
4 |
|
nfv |
⊢ Ⅎ 𝑟 ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) |
5 |
|
nfcv |
⊢ Ⅎ 𝑟 𝐴 |
6 |
|
nfra2w |
⊢ Ⅎ 𝑟 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) |
7 |
5 6
|
nfralw |
⊢ Ⅎ 𝑟 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) |
8 |
7
|
nfn |
⊢ Ⅎ 𝑟 ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) |
9 |
4 8
|
nfan |
⊢ Ⅎ 𝑟 ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
10 |
|
tospos |
⊢ ( 𝐾 ∈ Toset → 𝐾 ∈ Poset ) |
11 |
|
posprs |
⊢ ( 𝐾 ∈ Poset → 𝐾 ∈ Proset ) |
12 |
|
fvex |
⊢ ( le ‘ 𝐾 ) ∈ V |
13 |
12
|
inex1 |
⊢ ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) ∈ V |
14 |
2 13
|
eqeltri |
⊢ ≤ ∈ V |
15 |
|
eqid |
⊢ dom ≤ = dom ≤ |
16 |
15
|
ordttopon |
⊢ ( ≤ ∈ V → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ dom ≤ ) ) |
17 |
14 16
|
ax-mp |
⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ dom ≤ ) |
18 |
1 2
|
prsdm |
⊢ ( 𝐾 ∈ Proset → dom ≤ = 𝐵 ) |
19 |
18
|
fveq2d |
⊢ ( 𝐾 ∈ Proset → ( TopOn ‘ dom ≤ ) = ( TopOn ‘ 𝐵 ) ) |
20 |
17 19
|
eleqtrid |
⊢ ( 𝐾 ∈ Proset → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ 𝐵 ) ) |
21 |
3 20
|
eqeltrid |
⊢ ( 𝐾 ∈ Proset → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
22 |
10 11 21
|
3syl |
⊢ ( 𝐾 ∈ Toset → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
23 |
22
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
24 |
23
|
adantlr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
25 |
|
simpllr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐴 ⊆ 𝐵 ) |
26 |
25
|
adantlr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐴 ⊆ 𝐵 ) |
27 |
|
simpll |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → 𝐾 ∈ Toset ) |
28 |
27 10 11
|
3syl |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → 𝐾 ∈ Proset ) |
29 |
|
snex |
⊢ { 𝐵 } ∈ V |
30 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
31 |
30
|
mptex |
⊢ ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∈ V |
32 |
31
|
rnex |
⊢ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∈ V |
33 |
30
|
mptex |
⊢ ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ∈ V |
34 |
33
|
rnex |
⊢ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ∈ V |
35 |
32 34
|
unex |
⊢ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ∈ V |
36 |
29 35
|
unex |
⊢ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ∈ V |
37 |
|
ssfii |
⊢ ( ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ∈ V → ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ⊆ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) |
38 |
36 37
|
ax-mp |
⊢ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ⊆ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) |
39 |
|
fvex |
⊢ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ∈ V |
40 |
|
bastg |
⊢ ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ∈ V → ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ⊆ ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) ) |
41 |
39 40
|
ax-mp |
⊢ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ⊆ ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) |
42 |
38 41
|
sstri |
⊢ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ⊆ ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) |
43 |
|
eqid |
⊢ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) = ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) |
44 |
|
eqid |
⊢ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) = ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) |
45 |
1 2 43 44
|
ordtprsval |
⊢ ( 𝐾 ∈ Proset → ( ordTop ‘ ≤ ) = ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) ) |
46 |
3 45
|
syl5eq |
⊢ ( 𝐾 ∈ Proset → 𝐽 = ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ) ) ) |
47 |
42 46
|
sseqtrrid |
⊢ ( 𝐾 ∈ Proset → ( { 𝐵 } ∪ ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ) ⊆ 𝐽 ) |
48 |
47
|
unssbd |
⊢ ( 𝐾 ∈ Proset → ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ⊆ 𝐽 ) |
49 |
28 48
|
syl |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ( ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ∪ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) ⊆ 𝐽 ) |
50 |
49
|
unssbd |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ⊆ 𝐽 ) |
51 |
|
breq2 |
⊢ ( 𝑧 = 𝑦 → ( 𝑟 ≤ 𝑧 ↔ 𝑟 ≤ 𝑦 ) ) |
52 |
51
|
notbid |
⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑟 ≤ 𝑧 ↔ ¬ 𝑟 ≤ 𝑦 ) ) |
53 |
52
|
cbvrabv |
⊢ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑦 } |
54 |
|
breq1 |
⊢ ( 𝑥 = 𝑟 → ( 𝑥 ≤ 𝑦 ↔ 𝑟 ≤ 𝑦 ) ) |
55 |
54
|
notbid |
⊢ ( 𝑥 = 𝑟 → ( ¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑟 ≤ 𝑦 ) ) |
56 |
55
|
rabbidv |
⊢ ( 𝑥 = 𝑟 → { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑦 } ) |
57 |
56
|
rspceeqv |
⊢ ( ( 𝑟 ∈ 𝐵 ∧ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑦 } ) → ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) |
58 |
53 57
|
mpan2 |
⊢ ( 𝑟 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) |
59 |
30
|
rabex |
⊢ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ V |
60 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) = ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) |
61 |
60
|
elrnmpt |
⊢ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ V → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ↔ ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) |
62 |
59 61
|
ax-mp |
⊢ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ↔ ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) |
63 |
58 62
|
sylibr |
⊢ ( 𝑟 ∈ 𝐵 → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) |
64 |
63
|
adantl |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦 } ) ) |
65 |
50 64
|
sseldd |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ 𝐽 ) |
66 |
65
|
ad2antrr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∈ 𝐽 ) |
67 |
49
|
unssad |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ⊆ 𝐽 ) |
68 |
|
breq1 |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 ≤ 𝑟 ↔ 𝑦 ≤ 𝑟 ) ) |
69 |
68
|
notbid |
⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑧 ≤ 𝑟 ↔ ¬ 𝑦 ≤ 𝑟 ) ) |
70 |
69
|
cbvrabv |
⊢ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑟 } |
71 |
|
breq2 |
⊢ ( 𝑥 = 𝑟 → ( 𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑟 ) ) |
72 |
71
|
notbid |
⊢ ( 𝑥 = 𝑟 → ( ¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦 ≤ 𝑟 ) ) |
73 |
72
|
rabbidv |
⊢ ( 𝑥 = 𝑟 → { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑟 } ) |
74 |
73
|
rspceeqv |
⊢ ( ( 𝑟 ∈ 𝐵 ∧ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑟 } ) → ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) |
75 |
70 74
|
mpan2 |
⊢ ( 𝑟 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) |
76 |
30
|
rabex |
⊢ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ V |
77 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) = ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) |
78 |
77
|
elrnmpt |
⊢ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ V → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ↔ ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ) |
79 |
76 78
|
ax-mp |
⊢ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ↔ ∃ 𝑥 ∈ 𝐵 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } = { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) |
80 |
75 79
|
sylibr |
⊢ ( 𝑟 ∈ 𝐵 → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ) |
81 |
80
|
adantl |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ ran ( 𝑥 ∈ 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥 } ) ) |
82 |
67 81
|
sseldd |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ 𝐽 ) |
83 |
82
|
ad2antrr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∈ 𝐽 ) |
84 |
|
simpll |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ) |
85 |
|
simpr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ¬ 𝑟 ∈ 𝐴 ) |
86 |
84 85
|
jca |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
87 |
|
simplrl |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ) |
88 |
|
ssel |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
89 |
88
|
ancrd |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ) ) |
90 |
89
|
anim1d |
⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑥 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) ) ) |
91 |
90
|
impl |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) ) |
92 |
|
elin |
⊢ ( 𝑥 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ↔ ( 𝑥 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∧ 𝑥 ∈ 𝐴 ) ) |
93 |
|
breq2 |
⊢ ( 𝑧 = 𝑥 → ( 𝑟 ≤ 𝑧 ↔ 𝑟 ≤ 𝑥 ) ) |
94 |
93
|
notbid |
⊢ ( 𝑧 = 𝑥 → ( ¬ 𝑟 ≤ 𝑧 ↔ ¬ 𝑟 ≤ 𝑥 ) ) |
95 |
94
|
elrab |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑥 ) ) |
96 |
95
|
anbi1i |
⊢ ( ( 𝑥 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∧ 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑥 ) ∧ 𝑥 ∈ 𝐴 ) ) |
97 |
|
an32 |
⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑥 ) ∧ 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) ) |
98 |
92 96 97
|
3bitri |
⊢ ( 𝑥 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) ) |
99 |
91 98
|
sylibr |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) → 𝑥 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ) |
100 |
99
|
ne0d |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ≠ ∅ ) |
101 |
25 100
|
sylanl1 |
⊢ ( ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ 𝑥 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ≠ ∅ ) |
102 |
101
|
r19.29an |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ≠ ∅ ) |
103 |
86 87 102
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ 𝐴 ) ≠ ∅ ) |
104 |
|
simplrr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) |
105 |
|
ssel |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵 ) ) |
106 |
105
|
ancrd |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑦 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ) ) |
107 |
106
|
anim1d |
⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑟 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) ) ) |
108 |
107
|
impl |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) ) |
109 |
|
elin |
⊢ ( 𝑦 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ↔ ( 𝑦 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∧ 𝑦 ∈ 𝐴 ) ) |
110 |
69
|
elrab |
⊢ ( 𝑦 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ↔ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ≤ 𝑟 ) ) |
111 |
110
|
anbi1i |
⊢ ( ( 𝑦 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∧ 𝑦 ∈ 𝐴 ) ↔ ( ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ≤ 𝑟 ) ∧ 𝑦 ∈ 𝐴 ) ) |
112 |
|
an32 |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ≤ 𝑟 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) ) |
113 |
109 111 112
|
3bitri |
⊢ ( 𝑦 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ↔ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) ) |
114 |
108 113
|
sylibr |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) → 𝑦 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ) |
115 |
114
|
ne0d |
⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ≠ ∅ ) |
116 |
25 115
|
sylanl1 |
⊢ ( ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ≤ 𝑟 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ≠ ∅ ) |
117 |
116
|
r19.29an |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ≠ ∅ ) |
118 |
86 104 117
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ∩ 𝐴 ) ≠ ∅ ) |
119 |
1 2
|
trleile |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑟 ≤ 𝑧 ∨ 𝑧 ≤ 𝑟 ) ) |
120 |
|
oran |
⊢ ( ( 𝑟 ≤ 𝑧 ∨ 𝑧 ≤ 𝑟 ) ↔ ¬ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
121 |
119 120
|
sylib |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ¬ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
122 |
121
|
3expa |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐵 ) → ¬ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
123 |
122
|
nrexdv |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ) → ¬ ∃ 𝑧 ∈ 𝐵 ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
124 |
|
rabid |
⊢ ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ↔ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑧 ) ) |
125 |
|
rabid |
⊢ ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ↔ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
126 |
124 125
|
anbi12i |
⊢ ( ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∧ 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑧 ) ∧ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
127 |
|
elin |
⊢ ( 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∧ 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
128 |
|
anandi |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑧 ) ∧ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
129 |
126 127 128
|
3bitr4i |
⊢ ( 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
130 |
129
|
exbii |
⊢ ( ∃ 𝑧 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
131 |
|
nfrab1 |
⊢ Ⅎ 𝑧 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } |
132 |
|
nfrab1 |
⊢ Ⅎ 𝑧 { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } |
133 |
131 132
|
nfin |
⊢ Ⅎ 𝑧 ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) |
134 |
133
|
n0f |
⊢ ( ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ≠ ∅ ↔ ∃ 𝑧 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
135 |
|
df-rex |
⊢ ( ∃ 𝑧 ∈ 𝐵 ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
136 |
130 134 135
|
3bitr4i |
⊢ ( ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ≠ ∅ ↔ ∃ 𝑧 ∈ 𝐵 ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ) |
137 |
136
|
necon1bbii |
⊢ ( ¬ ∃ 𝑧 ∈ 𝐵 ( ¬ 𝑟 ≤ 𝑧 ∧ ¬ 𝑧 ≤ 𝑟 ) ↔ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ∅ ) |
138 |
123 137
|
sylib |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ∅ ) |
139 |
138
|
adantlr |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ∅ ) |
140 |
139
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ∅ ) |
141 |
140
|
ineq1d |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ∩ 𝐴 ) = ( ∅ ∩ 𝐴 ) ) |
142 |
|
0in |
⊢ ( ∅ ∩ 𝐴 ) = ∅ |
143 |
141 142
|
eqtrdi |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ∩ 𝐴 ) = ∅ ) |
144 |
143
|
adantlr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∩ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ∩ 𝐴 ) = ∅ ) |
145 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝑟 ∈ 𝐵 ) |
146 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ¬ 𝑟 ∈ 𝐴 ) |
147 |
|
vex |
⊢ 𝑟 ∈ V |
148 |
147
|
snss |
⊢ ( 𝑟 ∈ 𝐵 ↔ { 𝑟 } ⊆ 𝐵 ) |
149 |
|
eldif |
⊢ ( 𝑟 ∈ ( 𝐵 ∖ 𝐴 ) ↔ ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
150 |
147
|
snss |
⊢ ( 𝑟 ∈ ( 𝐵 ∖ 𝐴 ) ↔ { 𝑟 } ⊆ ( 𝐵 ∖ 𝐴 ) ) |
151 |
149 150
|
bitr3i |
⊢ ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ { 𝑟 } ⊆ ( 𝐵 ∖ 𝐴 ) ) |
152 |
|
ssconb |
⊢ ( ( { 𝑟 } ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵 ) → ( { 𝑟 } ⊆ ( 𝐵 ∖ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
153 |
151 152
|
syl5bb |
⊢ ( ( { 𝑟 } ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
154 |
148 153
|
sylanb |
⊢ ( ( 𝑟 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
155 |
154
|
adantl |
⊢ ( ( 𝐾 ∈ Toset ∧ ( 𝑟 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
156 |
155
|
anass1rs |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
157 |
156
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( 𝑟 ∈ 𝐵 ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) ) |
158 |
145 146 157
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐴 ⊆ ( 𝐵 ∖ { 𝑟 } ) ) |
159 |
10
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
160 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ) |
161 |
131 132
|
nfun |
⊢ Ⅎ 𝑧 ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) |
162 |
|
nfcv |
⊢ Ⅎ 𝑧 ( 𝐵 ∖ { 𝑟 } ) |
163 |
|
ianor |
⊢ ( ¬ ( 𝑟 ≤ 𝑧 ∧ 𝑧 ≤ 𝑟 ) ↔ ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ) |
164 |
1 2
|
posrasymb |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑟 ≤ 𝑧 ∧ 𝑧 ≤ 𝑟 ) ↔ 𝑟 = 𝑧 ) ) |
165 |
|
equcom |
⊢ ( 𝑟 = 𝑧 ↔ 𝑧 = 𝑟 ) |
166 |
164 165
|
syl6bb |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑟 ≤ 𝑧 ∧ 𝑧 ≤ 𝑟 ) ↔ 𝑧 = 𝑟 ) ) |
167 |
166
|
necon3bbid |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ¬ ( 𝑟 ≤ 𝑧 ∧ 𝑧 ≤ 𝑟 ) ↔ 𝑧 ≠ 𝑟 ) ) |
168 |
163 167
|
bitr3id |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ↔ 𝑧 ≠ 𝑟 ) ) |
169 |
168
|
3expia |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐵 → ( ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ↔ 𝑧 ≠ 𝑟 ) ) ) |
170 |
169
|
pm5.32d |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ) → ( ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑧 ≠ 𝑟 ) ) ) |
171 |
124 125
|
orbi12i |
⊢ ( ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∨ 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑧 ) ∨ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
172 |
|
elun |
⊢ ( 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ ( 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∨ 𝑧 ∈ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
173 |
|
andi |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑟 ≤ 𝑧 ) ∨ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ≤ 𝑟 ) ) ) |
174 |
171 172 173
|
3bitr4ri |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ ( ¬ 𝑟 ≤ 𝑧 ∨ ¬ 𝑧 ≤ 𝑟 ) ) ↔ 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
175 |
|
eldifsn |
⊢ ( 𝑧 ∈ ( 𝐵 ∖ { 𝑟 } ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑧 ≠ 𝑟 ) ) |
176 |
175
|
bicomi |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ 𝑧 ≠ 𝑟 ) ↔ 𝑧 ∈ ( 𝐵 ∖ { 𝑟 } ) ) |
177 |
170 174 176
|
3bitr3g |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ) → ( 𝑧 ∈ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ↔ 𝑧 ∈ ( 𝐵 ∖ { 𝑟 } ) ) ) |
178 |
160 161 162 177
|
eqrd |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ( 𝐵 ∖ { 𝑟 } ) ) |
179 |
159 145 178
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) = ( 𝐵 ∖ { 𝑟 } ) ) |
180 |
158 179
|
sseqtrrd |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐴 ⊆ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
181 |
180
|
adantlr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝐴 ⊆ ( { 𝑧 ∈ 𝐵 ∣ ¬ 𝑟 ≤ 𝑧 } ∪ { 𝑧 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑟 } ) ) |
182 |
24 26 66 83 103 118 144 181
|
nconnsubb |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ¬ ( 𝐽 ↾t 𝐴 ) ∈ Conn ) |
183 |
182
|
anasss |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) → ¬ ( 𝐽 ↾t 𝐴 ) ∈ Conn ) |
184 |
183
|
adantllr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) ∧ 𝑟 ∈ 𝐵 ) ∧ ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) → ¬ ( 𝐽 ↾t 𝐴 ) ∈ Conn ) |
185 |
|
rexanali |
⊢ ( ∃ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ¬ ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
186 |
185
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑦 ∈ 𝐴 ¬ ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
187 |
|
rexcom |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
188 |
|
rexnal |
⊢ ( ∃ 𝑦 ∈ 𝐴 ¬ ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ↔ ¬ ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
189 |
186 187 188
|
3bitr3i |
⊢ ( ∃ 𝑟 ∈ 𝐵 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ¬ ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
190 |
189
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑟 ∈ 𝐵 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
191 |
|
rexcom |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑟 ∈ 𝐵 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
192 |
|
rexnal |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
193 |
190 191 192
|
3bitr3i |
⊢ ( ∃ 𝑟 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) |
194 |
|
r19.41v |
⊢ ( ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ( ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
195 |
194
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
196 |
|
r19.41v |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
197 |
|
reeanv |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ) |
198 |
197
|
anbi1i |
⊢ ( ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
199 |
195 196 198
|
3bitri |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
200 |
199
|
rexbii |
⊢ ( ∃ 𝑟 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
201 |
193 200
|
bitr3i |
⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
202 |
27
|
ad2antrr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐾 ∈ Toset ) |
203 |
25
|
sselda |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐵 ) |
204 |
|
simpllr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑟 ∈ 𝐵 ) |
205 |
1 2
|
trleile |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑥 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵 ) → ( 𝑥 ≤ 𝑟 ∨ 𝑟 ≤ 𝑥 ) ) |
206 |
202 203 204 205
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ≤ 𝑟 ∨ 𝑟 ≤ 𝑥 ) ) |
207 |
|
simpr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
208 |
|
simplr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑟 ∈ 𝐴 ) |
209 |
|
nelne2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝑥 ≠ 𝑟 ) |
210 |
207 208 209
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≠ 𝑟 ) |
211 |
159
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
212 |
1 2
|
posrasymb |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵 ) → ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ↔ 𝑥 = 𝑟 ) ) |
213 |
212
|
necon3bbid |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵 ) → ( ¬ ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ↔ 𝑥 ≠ 𝑟 ) ) |
214 |
211 203 204 213
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( ¬ ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ↔ 𝑥 ≠ 𝑟 ) ) |
215 |
210 214
|
mpbird |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ¬ ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ) |
216 |
206 215
|
jca |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ≤ 𝑟 ∨ 𝑟 ≤ 𝑥 ) ∧ ¬ ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ) ) |
217 |
|
pm5.17 |
⊢ ( ( ( 𝑥 ≤ 𝑟 ∨ 𝑟 ≤ 𝑥 ) ∧ ¬ ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑥 ) ) ↔ ( 𝑥 ≤ 𝑟 ↔ ¬ 𝑟 ≤ 𝑥 ) ) |
218 |
216 217
|
sylib |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ≤ 𝑟 ↔ ¬ 𝑟 ≤ 𝑥 ) ) |
219 |
218
|
rexbidva |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ↔ ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ) ) |
220 |
27
|
ad2antrr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝐾 ∈ Toset ) |
221 |
|
simpllr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑟 ∈ 𝐵 ) |
222 |
25
|
sselda |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) |
223 |
1 2
|
trleile |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑟 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑟 ≤ 𝑦 ∨ 𝑦 ≤ 𝑟 ) ) |
224 |
220 221 222 223
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑟 ≤ 𝑦 ∨ 𝑦 ≤ 𝑟 ) ) |
225 |
|
simpr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) |
226 |
|
simplr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ¬ 𝑟 ∈ 𝐴 ) |
227 |
|
nelne2 |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑟 ∈ 𝐴 ) → 𝑦 ≠ 𝑟 ) |
228 |
225 226 227
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≠ 𝑟 ) |
229 |
228
|
necomd |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑟 ≠ 𝑦 ) |
230 |
159
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
231 |
1 2
|
posrasymb |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ↔ 𝑟 = 𝑦 ) ) |
232 |
231
|
necon3bbid |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑟 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ¬ ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ↔ 𝑟 ≠ 𝑦 ) ) |
233 |
230 221 222 232
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ¬ ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ↔ 𝑟 ≠ 𝑦 ) ) |
234 |
229 233
|
mpbird |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ¬ ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ) |
235 |
224 234
|
jca |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑟 ≤ 𝑦 ∨ 𝑦 ≤ 𝑟 ) ∧ ¬ ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ) ) |
236 |
|
pm5.17 |
⊢ ( ( ( 𝑟 ≤ 𝑦 ∨ 𝑦 ≤ 𝑟 ) ∧ ¬ ( 𝑟 ≤ 𝑦 ∧ 𝑦 ≤ 𝑟 ) ) ↔ ( 𝑟 ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑟 ) ) |
237 |
235 236
|
sylib |
⊢ ( ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑟 ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑟 ) ) |
238 |
237
|
rexbidva |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ↔ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) |
239 |
219 238
|
anbi12d |
⊢ ( ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) ∧ ¬ 𝑟 ∈ 𝐴 ) → ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ) |
240 |
239
|
ex |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ( ¬ 𝑟 ∈ 𝐴 → ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ) ) ) |
241 |
240
|
pm5.32rd |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝑟 ∈ 𝐵 ) → ( ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) ) |
242 |
241
|
rexbidva |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) → ( ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑟 ∧ ∃ 𝑦 ∈ 𝐴 𝑟 ≤ 𝑦 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) ) |
243 |
201 242
|
syl5bb |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) → ( ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ↔ ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) ) |
244 |
243
|
biimpa |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) → ∃ 𝑟 ∈ 𝐵 ( ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑟 ≤ 𝑥 ∧ ∃ 𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑟 ) ∧ ¬ 𝑟 ∈ 𝐴 ) ) |
245 |
9 184 244
|
r19.29af |
⊢ ( ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) ∧ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) → ¬ ( 𝐽 ↾t 𝐴 ) ∈ Conn ) |
246 |
245
|
ex |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) → ( ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) → ¬ ( 𝐽 ↾t 𝐴 ) ∈ Conn ) ) |
247 |
246
|
con4d |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝐽 ↾t 𝐴 ) ∈ Conn → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑟 ∈ 𝐵 ( ( 𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦 ) → 𝑟 ∈ 𝐴 ) ) ) |