Step |
Hyp |
Ref |
Expression |
1 |
|
pospo.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pospo.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pospo.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
3
|
pltirr |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ) → ¬ 𝑥 < 𝑥 ) |
5 |
1 3
|
plttr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 < 𝑧 ) ) |
6 |
4 5
|
ispod |
⊢ ( 𝐾 ∈ Poset → < Po 𝐵 ) |
7 |
|
relres |
⊢ Rel ( I ↾ 𝐵 ) |
8 |
7
|
a1i |
⊢ ( 𝐾 ∈ Poset → Rel ( I ↾ 𝐵 ) ) |
9 |
|
opabresid |
⊢ ( I ↾ 𝐵 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } |
10 |
9
|
eqcomi |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } = ( I ↾ 𝐵 ) |
11 |
10
|
eleq2i |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } ↔ 〈 𝑥 , 𝑦 〉 ∈ ( I ↾ 𝐵 ) ) |
12 |
|
opabidw |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) } ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ) |
13 |
11 12
|
bitr3i |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( I ↾ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) ) |
14 |
1 2
|
posref |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ≤ 𝑥 ) |
15 |
|
df-br |
⊢ ( 𝑥 ≤ 𝑦 ↔ 〈 𝑥 , 𝑦 〉 ∈ ≤ ) |
16 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑥 ) ) |
17 |
15 16
|
bitr3id |
⊢ ( 𝑦 = 𝑥 → ( 〈 𝑥 , 𝑦 〉 ∈ ≤ ↔ 𝑥 ≤ 𝑥 ) ) |
18 |
14 17
|
syl5ibrcom |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 = 𝑥 → 〈 𝑥 , 𝑦 〉 ∈ ≤ ) ) |
19 |
18
|
expimpd |
⊢ ( 𝐾 ∈ Poset → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥 ) → 〈 𝑥 , 𝑦 〉 ∈ ≤ ) ) |
20 |
13 19
|
syl5bi |
⊢ ( 𝐾 ∈ Poset → ( 〈 𝑥 , 𝑦 〉 ∈ ( I ↾ 𝐵 ) → 〈 𝑥 , 𝑦 〉 ∈ ≤ ) ) |
21 |
8 20
|
relssdv |
⊢ ( 𝐾 ∈ Poset → ( I ↾ 𝐵 ) ⊆ ≤ ) |
22 |
6 21
|
jca |
⊢ ( 𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) |
23 |
|
simpl |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) → 𝐾 ∈ 𝑉 ) |
24 |
1
|
a1i |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) → 𝐵 = ( Base ‘ 𝐾 ) ) |
25 |
2
|
a1i |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) → ≤ = ( le ‘ 𝐾 ) ) |
26 |
|
equid |
⊢ 𝑥 = 𝑥 |
27 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
28 |
|
resieq |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( I ↾ 𝐵 ) 𝑥 ↔ 𝑥 = 𝑥 ) ) |
29 |
27 27 28
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( I ↾ 𝐵 ) 𝑥 ↔ 𝑥 = 𝑥 ) ) |
30 |
26 29
|
mpbiri |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ( I ↾ 𝐵 ) 𝑥 ) |
31 |
|
simplrr |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → ( I ↾ 𝐵 ) ⊆ ≤ ) |
32 |
31
|
ssbrd |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( I ↾ 𝐵 ) 𝑥 → 𝑥 ≤ 𝑥 ) ) |
33 |
30 32
|
mpd |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ≤ 𝑥 ) |
34 |
1 2 3
|
pleval2i |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ≤ 𝑦 → ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
35 |
34
|
3adant1 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ≤ 𝑦 → ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
36 |
1 2 3
|
pleval2i |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 → ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) |
37 |
36
|
ancoms |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 → ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) |
38 |
37
|
3adant1 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 → ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) |
39 |
|
simprl |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) → < Po 𝐵 ) |
40 |
|
po2nr |
⊢ ( ( < Po 𝐵 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ¬ ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑥 ) ) |
41 |
40
|
3impb |
⊢ ( ( < Po 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ¬ ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑥 ) ) |
42 |
39 41
|
syl3an1 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ¬ ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑥 ) ) |
43 |
42
|
pm2.21d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑥 ) → 𝑥 = 𝑦 ) ) |
44 |
|
simpl |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑦 < 𝑥 ) → 𝑥 = 𝑦 ) |
45 |
44
|
a1i |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 = 𝑦 ∧ 𝑦 < 𝑥 ) → 𝑥 = 𝑦 ) ) |
46 |
|
simpr |
⊢ ( ( 𝑥 < 𝑦 ∧ 𝑦 = 𝑥 ) → 𝑦 = 𝑥 ) |
47 |
46
|
equcomd |
⊢ ( ( 𝑥 < 𝑦 ∧ 𝑦 = 𝑥 ) → 𝑥 = 𝑦 ) |
48 |
47
|
a1i |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 = 𝑥 ) → 𝑥 = 𝑦 ) ) |
49 |
|
simpl |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑦 = 𝑥 ) → 𝑥 = 𝑦 ) |
50 |
49
|
a1i |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 = 𝑦 ∧ 𝑦 = 𝑥 ) → 𝑥 = 𝑦 ) ) |
51 |
43 45 48 50
|
ccased |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∧ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) → 𝑥 = 𝑦 ) ) |
52 |
35 38 51
|
syl2and |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥 ) → 𝑥 = 𝑦 ) ) |
53 |
|
simpr1 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
54 |
|
simpr2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
55 |
53 54 34
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ≤ 𝑦 → ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
56 |
|
simpr3 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑧 ∈ 𝐵 ) |
57 |
1 2 3
|
pleval2i |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑧 → ( 𝑦 < 𝑧 ∨ 𝑦 = 𝑧 ) ) ) |
58 |
54 56 57
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ≤ 𝑧 → ( 𝑦 < 𝑧 ∨ 𝑦 = 𝑧 ) ) ) |
59 |
|
potr |
⊢ ( ( < Po 𝐵 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 < 𝑧 ) ) |
60 |
39 59
|
sylan |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 < 𝑧 ) ) |
61 |
|
simpll |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝐾 ∈ 𝑉 ) |
62 |
2 3
|
pltle |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑥 < 𝑧 → 𝑥 ≤ 𝑧 ) ) |
63 |
61 53 56 62
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 < 𝑧 → 𝑥 ≤ 𝑧 ) ) |
64 |
60 63
|
syld |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 ≤ 𝑧 ) ) |
65 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 𝑧 ↔ 𝑦 < 𝑧 ) ) |
66 |
65
|
biimpar |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 < 𝑧 ) |
67 |
66 63
|
syl5 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 = 𝑦 ∧ 𝑦 < 𝑧 ) → 𝑥 ≤ 𝑧 ) ) |
68 |
|
breq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 < 𝑦 ↔ 𝑥 < 𝑧 ) ) |
69 |
68
|
biimpac |
⊢ ( ( 𝑥 < 𝑦 ∧ 𝑦 = 𝑧 ) → 𝑥 < 𝑧 ) |
70 |
69 63
|
syl5 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 < 𝑦 ∧ 𝑦 = 𝑧 ) → 𝑥 ≤ 𝑧 ) ) |
71 |
53 33
|
syldan |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑥 ≤ 𝑥 ) |
72 |
|
eqtr |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑦 = 𝑧 ) → 𝑥 = 𝑧 ) |
73 |
72
|
breq2d |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑦 = 𝑧 ) → ( 𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑧 ) ) |
74 |
71 73
|
syl5ibcom |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 = 𝑦 ∧ 𝑦 = 𝑧 ) → 𝑥 ≤ 𝑧 ) ) |
75 |
64 67 70 74
|
ccased |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∧ ( 𝑦 < 𝑧 ∨ 𝑦 = 𝑧 ) ) → 𝑥 ≤ 𝑧 ) ) |
76 |
55 58 75
|
syl2and |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) |
77 |
23 24 25 33 52 76
|
isposd |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) → 𝐾 ∈ Poset ) |
78 |
77
|
ex |
⊢ ( 𝐾 ∈ 𝑉 → ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) → 𝐾 ∈ Poset ) ) |
79 |
22 78
|
impbid2 |
⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ) |