Step |
Hyp |
Ref |
Expression |
1 |
|
0sno |
⊢ 0s ∈ No |
2 |
|
slerflex |
⊢ ( 0s ∈ No → 0s ≤s 0s ) |
3 |
1 2
|
ax-mp |
⊢ 0s ≤s 0s |
4 |
1
|
elexi |
⊢ 0s ∈ V |
5 |
|
breq2 |
⊢ ( 𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ) ) |
6 |
4 5
|
rexsn |
⊢ ( ∃ 𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s ) |
7 |
3 6
|
mpbir |
⊢ ∃ 𝑥 ∈ { 0s } 0s ≤s 𝑥 |
8 |
7
|
orci |
⊢ ( ∃ 𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃ 𝑦 ∈ ∅ 𝑦 ≤s 1s ) |
9 |
|
0elpw |
⊢ ∅ ∈ 𝒫 No |
10 |
|
nulssgt |
⊢ ( ∅ ∈ 𝒫 No → ∅ <<s ∅ ) |
11 |
9 10
|
ax-mp |
⊢ ∅ <<s ∅ |
12 |
|
snssi |
⊢ ( 0s ∈ No → { 0s } ⊆ No ) |
13 |
1 12
|
ax-mp |
⊢ { 0s } ⊆ No |
14 |
|
snex |
⊢ { 0s } ∈ V |
15 |
14
|
elpw |
⊢ ( { 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) |
16 |
13 15
|
mpbir |
⊢ { 0s } ∈ 𝒫 No |
17 |
|
nulssgt |
⊢ ( { 0s } ∈ 𝒫 No → { 0s } <<s ∅ ) |
18 |
16 17
|
ax-mp |
⊢ { 0s } <<s ∅ |
19 |
|
df-0s |
⊢ 0s = ( ∅ |s ∅ ) |
20 |
|
df-1s |
⊢ 1s = ( { 0s } |s ∅ ) |
21 |
|
sltrec |
⊢ ( ( ( ∅ <<s ∅ ∧ { 0s } <<s ∅ ) ∧ ( 0s = ( ∅ |s ∅ ) ∧ 1s = ( { 0s } |s ∅ ) ) ) → ( 0s <s 1s ↔ ( ∃ 𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃ 𝑦 ∈ ∅ 𝑦 ≤s 1s ) ) ) |
22 |
11 18 19 20 21
|
mp4an |
⊢ ( 0s <s 1s ↔ ( ∃ 𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃ 𝑦 ∈ ∅ 𝑦 ≤s 1s ) ) |
23 |
8 22
|
mpbir |
⊢ 0s <s 1s |