Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → 𝐴 ≠ ∅ ) |
2 |
|
relen |
⊢ Rel ≈ |
3 |
2
|
brrelex1i |
⊢ ( 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) → 𝐴 ∈ V ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → 𝐴 ∈ V ) |
5 |
|
0sdomg |
⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
6 |
4 5
|
syl |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
7 |
1 6
|
mpbird |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ∅ ≺ 𝐴 ) |
8 |
|
0sdom1dom |
⊢ ( ∅ ≺ 𝐴 ↔ 1o ≼ 𝐴 ) |
9 |
7 8
|
sylib |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → 1o ≼ 𝐴 ) |
10 |
|
djudom2 |
⊢ ( ( 1o ≼ 𝐴 ∧ 𝐴 ∈ V ) → ( 𝐴 ⊔ 1o ) ≼ ( 𝐴 ⊔ 𝐴 ) ) |
11 |
9 4 10
|
syl2anc |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ( 𝐴 ⊔ 1o ) ≼ ( 𝐴 ⊔ 𝐴 ) ) |
12 |
|
domen2 |
⊢ ( 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) → ( ( 𝐴 ⊔ 1o ) ≼ 𝐴 ↔ ( 𝐴 ⊔ 1o ) ≼ ( 𝐴 ⊔ 𝐴 ) ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ( ( 𝐴 ⊔ 1o ) ≼ 𝐴 ↔ ( 𝐴 ⊔ 1o ) ≼ ( 𝐴 ⊔ 𝐴 ) ) ) |
14 |
11 13
|
mpbird |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ( 𝐴 ⊔ 1o ) ≼ 𝐴 ) |
15 |
|
domnsym |
⊢ ( ( 𝐴 ⊔ 1o ) ≼ 𝐴 → ¬ 𝐴 ≺ ( 𝐴 ⊔ 1o ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) → ¬ 𝐴 ≺ ( 𝐴 ⊔ 1o ) ) |
17 |
|
isfin4p1 |
⊢ ( 𝐴 ∈ FinIV ↔ 𝐴 ≺ ( 𝐴 ⊔ 1o ) ) |
18 |
17
|
biimpi |
⊢ ( 𝐴 ∈ FinIV → 𝐴 ≺ ( 𝐴 ⊔ 1o ) ) |
19 |
16 18
|
nsyl3 |
⊢ ( 𝐴 ∈ FinIV → ¬ ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) ) |
20 |
|
isfin5-2 |
⊢ ( 𝐴 ∈ FinIV → ( 𝐴 ∈ FinV ↔ ¬ ( 𝐴 ≠ ∅ ∧ 𝐴 ≈ ( 𝐴 ⊔ 𝐴 ) ) ) ) |
21 |
19 20
|
mpbird |
⊢ ( 𝐴 ∈ FinIV → 𝐴 ∈ FinV ) |