Step |
Hyp |
Ref |
Expression |
1 |
|
pweq |
⊢ ( 𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅ ) |
2 |
1
|
breq1d |
⊢ ( 𝐴 = ∅ → ( 𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵 ) ) |
3 |
|
reldom |
⊢ Rel ≼ |
4 |
3
|
brrelex1i |
⊢ ( 𝐴 ≼ 𝐵 → 𝐴 ∈ V ) |
5 |
|
0sdomg |
⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ≼ 𝐵 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
7 |
6
|
biimpar |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅ ) → ∅ ≺ 𝐴 ) |
8 |
|
simpl |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅ ) → 𝐴 ≼ 𝐵 ) |
9 |
|
fodomr |
⊢ ( ( ∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) |
10 |
7 8 9
|
syl2anc |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) |
11 |
|
vex |
⊢ 𝑓 ∈ V |
12 |
|
fopwdom |
⊢ ( ( 𝑓 ∈ V ∧ 𝑓 : 𝐵 –onto→ 𝐴 ) → 𝒫 𝐴 ≼ 𝒫 𝐵 ) |
13 |
11 12
|
mpan |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵 ) |
14 |
13
|
exlimiv |
⊢ ( ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵 ) |
15 |
10 14
|
syl |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅ ) → 𝒫 𝐴 ≼ 𝒫 𝐵 ) |
16 |
3
|
brrelex2i |
⊢ ( 𝐴 ≼ 𝐵 → 𝐵 ∈ V ) |
17 |
16
|
pwexd |
⊢ ( 𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V ) |
18 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
19 |
18
|
sspwi |
⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
20 |
|
ssdomg |
⊢ ( 𝒫 𝐵 ∈ V → ( 𝒫 ∅ ⊆ 𝒫 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵 ) ) |
21 |
17 19 20
|
mpisyl |
⊢ ( 𝐴 ≼ 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵 ) |
22 |
2 15 21
|
pm2.61ne |
⊢ ( 𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵 ) |