| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sltsex1 |
⊢ ( 𝐴 <<s 𝐶 → 𝐴 ∈ V ) |
| 2 |
1
|
3ad2ant3 |
⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 ∈ V ) |
| 3 |
|
simp1 |
⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐵 ∈ 𝒫 No ) |
| 4 |
|
sltsss1 |
⊢ ( 𝐴 <<s 𝐶 → 𝐴 ⊆ No ) |
| 5 |
4
|
3ad2ant3 |
⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 ⊆ No ) |
| 6 |
3
|
elpwid |
⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐵 ⊆ No ) |
| 7 |
|
breq2 |
⊢ ( 𝑥 = 𝑏 → ( 𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑏 ) ) |
| 8 |
7
|
rexbidv |
⊢ ( 𝑥 = 𝑏 → ( ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ↔ ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 ) ) |
| 9 |
|
simp12 |
⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ) |
| 10 |
|
simp3 |
⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ 𝐵 ) |
| 11 |
8 9 10
|
rspcdva |
⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 ) |
| 12 |
|
breq1 |
⊢ ( 𝑦 = 𝑐 → ( 𝑦 ≤s 𝑏 ↔ 𝑐 ≤s 𝑏 ) ) |
| 13 |
12
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 ↔ ∃ 𝑐 ∈ 𝐶 𝑐 ≤s 𝑏 ) |
| 14 |
11 13
|
sylib |
⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∃ 𝑐 ∈ 𝐶 𝑐 ≤s 𝑏 ) |
| 15 |
|
simpl13 |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐴 <<s 𝐶 ) |
| 16 |
15 4
|
syl |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐴 ⊆ No ) |
| 17 |
|
simpl2 |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 ∈ 𝐴 ) |
| 18 |
16 17
|
sseldd |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 ∈ No ) |
| 19 |
|
sltsss2 |
⊢ ( 𝐴 <<s 𝐶 → 𝐶 ⊆ No ) |
| 20 |
15 19
|
syl |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐶 ⊆ No ) |
| 21 |
|
simprl |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ∈ 𝐶 ) |
| 22 |
20 21
|
sseldd |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ∈ No ) |
| 23 |
|
simpl1 |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ) |
| 24 |
23 6
|
syl |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐵 ⊆ No ) |
| 25 |
|
simpl3 |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑏 ∈ 𝐵 ) |
| 26 |
24 25
|
sseldd |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑏 ∈ No ) |
| 27 |
15 17 21
|
sltssepcd |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 <s 𝑐 ) |
| 28 |
|
simprr |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ≤s 𝑏 ) |
| 29 |
18 22 26 27 28
|
ltlestrd |
⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 <s 𝑏 ) |
| 30 |
14 29
|
rexlimddv |
⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → 𝑎 <s 𝑏 ) |
| 31 |
2 3 5 6 30
|
sltsd |
⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 <<s 𝐵 ) |