Step |
Hyp |
Ref |
Expression |
1 |
|
ensn1g |
⊢ ( 𝐴 ∈ V → { 𝐴 } ≈ 1o ) |
2 |
|
1on |
⊢ 1o ∈ On |
3 |
|
domrefg |
⊢ ( 1o ∈ On → 1o ≼ 1o ) |
4 |
2 3
|
ax-mp |
⊢ 1o ≼ 1o |
5 |
|
endomtr |
⊢ ( ( { 𝐴 } ≈ 1o ∧ 1o ≼ 1o ) → { 𝐴 } ≼ 1o ) |
6 |
1 4 5
|
sylancl |
⊢ ( 𝐴 ∈ V → { 𝐴 } ≼ 1o ) |
7 |
|
snprc |
⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ ) |
8 |
|
snex |
⊢ { 𝐴 } ∈ V |
9 |
|
eqeng |
⊢ ( { 𝐴 } ∈ V → ( { 𝐴 } = ∅ → { 𝐴 } ≈ ∅ ) ) |
10 |
8 9
|
ax-mp |
⊢ ( { 𝐴 } = ∅ → { 𝐴 } ≈ ∅ ) |
11 |
7 10
|
sylbi |
⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } ≈ ∅ ) |
12 |
|
0domg |
⊢ ( 1o ∈ On → ∅ ≼ 1o ) |
13 |
2 12
|
ax-mp |
⊢ ∅ ≼ 1o |
14 |
|
endomtr |
⊢ ( ( { 𝐴 } ≈ ∅ ∧ ∅ ≼ 1o ) → { 𝐴 } ≼ 1o ) |
15 |
11 13 14
|
sylancl |
⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } ≼ 1o ) |
16 |
6 15
|
pm2.61i |
⊢ { 𝐴 } ≼ 1o |