Step |
Hyp |
Ref |
Expression |
1 |
|
en1 |
⊢ ( 𝐴 ≈ 1o ↔ ∃ 𝑥 𝐴 = { 𝑥 } ) |
2 |
|
id |
⊢ ( 𝐴 = { 𝑥 } → 𝐴 = { 𝑥 } ) |
3 |
|
unieq |
⊢ ( 𝐴 = { 𝑥 } → ∪ 𝐴 = ∪ { 𝑥 } ) |
4 |
|
unisnv |
⊢ ∪ { 𝑥 } = 𝑥 |
5 |
3 4
|
eqtrdi |
⊢ ( 𝐴 = { 𝑥 } → ∪ 𝐴 = 𝑥 ) |
6 |
5
|
sneqd |
⊢ ( 𝐴 = { 𝑥 } → { ∪ 𝐴 } = { 𝑥 } ) |
7 |
2 6
|
eqtr4d |
⊢ ( 𝐴 = { 𝑥 } → 𝐴 = { ∪ 𝐴 } ) |
8 |
7
|
exlimiv |
⊢ ( ∃ 𝑥 𝐴 = { 𝑥 } → 𝐴 = { ∪ 𝐴 } ) |
9 |
1 8
|
sylbi |
⊢ ( 𝐴 ≈ 1o → 𝐴 = { ∪ 𝐴 } ) |
10 |
|
id |
⊢ ( 𝐴 = { ∪ 𝐴 } → 𝐴 = { ∪ 𝐴 } ) |
11 |
|
eqsnuniex |
⊢ ( 𝐴 = { ∪ 𝐴 } → ∪ 𝐴 ∈ V ) |
12 |
|
ensn1g |
⊢ ( ∪ 𝐴 ∈ V → { ∪ 𝐴 } ≈ 1o ) |
13 |
11 12
|
syl |
⊢ ( 𝐴 = { ∪ 𝐴 } → { ∪ 𝐴 } ≈ 1o ) |
14 |
10 13
|
eqbrtrd |
⊢ ( 𝐴 = { ∪ 𝐴 } → 𝐴 ≈ 1o ) |
15 |
9 14
|
impbii |
⊢ ( 𝐴 ≈ 1o ↔ 𝐴 = { ∪ 𝐴 } ) |