| Step |
Hyp |
Ref |
Expression |
| 1 |
|
n0 |
⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝐵 ) |
| 2 |
|
ssltex1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ∈ V ) |
| 3 |
|
ssltex2 |
⊢ ( 𝐵 <<s 𝐶 → 𝐶 ∈ V ) |
| 4 |
2 3
|
anim12i |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) → ( 𝐴 ∈ V ∧ 𝐶 ∈ V ) ) |
| 5 |
4
|
adantl |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → ( 𝐴 ∈ V ∧ 𝐶 ∈ V ) ) |
| 6 |
|
ssltss1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
| 7 |
6
|
ad2antrl |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → 𝐴 ⊆ No ) |
| 8 |
|
ssltss2 |
⊢ ( 𝐵 <<s 𝐶 → 𝐶 ⊆ No ) |
| 9 |
8
|
ad2antll |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → 𝐶 ⊆ No ) |
| 10 |
7
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝐴 ⊆ No ) |
| 11 |
|
simprl |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 ∈ 𝐴 ) |
| 12 |
10 11
|
sseldd |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 ∈ No ) |
| 13 |
|
ssltss1 |
⊢ ( 𝐵 <<s 𝐶 → 𝐵 ⊆ No ) |
| 14 |
13
|
ad2antll |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → 𝐵 ⊆ No ) |
| 15 |
14
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝐵 ⊆ No ) |
| 16 |
|
simpll |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑦 ∈ 𝐵 ) |
| 17 |
15 16
|
sseldd |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑦 ∈ No ) |
| 18 |
9
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝐶 ⊆ No ) |
| 19 |
|
simprr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑧 ∈ 𝐶 ) |
| 20 |
18 19
|
sseldd |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑧 ∈ No ) |
| 21 |
|
ssltsep |
⊢ ( 𝐴 <<s 𝐵 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) |
| 22 |
21
|
ad2antrl |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) |
| 23 |
22
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) |
| 24 |
|
rsp |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) ) |
| 25 |
23 11 24
|
sylc |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) |
| 26 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ( 𝑦 ∈ 𝐵 → 𝑥 <s 𝑦 ) ) |
| 27 |
25 16 26
|
sylc |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 <s 𝑦 ) |
| 28 |
|
ssltsep |
⊢ ( 𝐵 <<s 𝐶 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 ) |
| 29 |
28
|
ad2antll |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 ) |
| 30 |
29
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 ) |
| 31 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 → ( 𝑦 ∈ 𝐵 → ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 ) ) |
| 32 |
30 16 31
|
sylc |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 ) |
| 33 |
|
rsp |
⊢ ( ∀ 𝑧 ∈ 𝐶 𝑦 <s 𝑧 → ( 𝑧 ∈ 𝐶 → 𝑦 <s 𝑧 ) ) |
| 34 |
32 19 33
|
sylc |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑦 <s 𝑧 ) |
| 35 |
12 17 20 27 34
|
slttrd |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶 ) ) → 𝑥 <s 𝑧 ) |
| 36 |
35
|
ralrimivva |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝑥 <s 𝑧 ) |
| 37 |
7 9 36
|
3jca |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → ( 𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝑥 <s 𝑧 ) ) |
| 38 |
|
brsslt |
⊢ ( 𝐴 <<s 𝐶 ↔ ( ( 𝐴 ∈ V ∧ 𝐶 ∈ V ) ∧ ( 𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝑥 <s 𝑧 ) ) ) |
| 39 |
5 37 38
|
sylanbrc |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) ) → 𝐴 <<s 𝐶 ) |
| 40 |
39
|
ex |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) → 𝐴 <<s 𝐶 ) ) |
| 41 |
40
|
exlimiv |
⊢ ( ∃ 𝑦 𝑦 ∈ 𝐵 → ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) → 𝐴 <<s 𝐶 ) ) |
| 42 |
1 41
|
sylbi |
⊢ ( 𝐵 ≠ ∅ → ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) → 𝐴 <<s 𝐶 ) ) |
| 43 |
42
|
com12 |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ) → ( 𝐵 ≠ ∅ → 𝐴 <<s 𝐶 ) ) |
| 44 |
43
|
3impia |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ∧ 𝐵 ≠ ∅ ) → 𝐴 <<s 𝐶 ) |