Step |
Hyp |
Ref |
Expression |
1 |
|
snssi |
⊢ ( 𝐴 ∈ Cℋ → { 𝐴 } ⊆ Cℋ ) |
2 |
|
chsupval2 |
⊢ ( { 𝐴 } ⊆ Cℋ → ( ∨ℋ ‘ { 𝐴 } ) = ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ) |
3 |
1 2
|
syl |
⊢ ( 𝐴 ∈ Cℋ → ( ∨ℋ ‘ { 𝐴 } ) = ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ) |
4 |
|
unisng |
⊢ ( 𝐴 ∈ Cℋ → ∪ { 𝐴 } = 𝐴 ) |
5 |
|
eqimss |
⊢ ( ∪ { 𝐴 } = 𝐴 → ∪ { 𝐴 } ⊆ 𝐴 ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ Cℋ → ∪ { 𝐴 } ⊆ 𝐴 ) |
7 |
6
|
ancli |
⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∈ Cℋ ∧ ∪ { 𝐴 } ⊆ 𝐴 ) ) |
8 |
|
sseq2 |
⊢ ( 𝑥 = 𝐴 → ( ∪ { 𝐴 } ⊆ 𝑥 ↔ ∪ { 𝐴 } ⊆ 𝐴 ) ) |
9 |
8
|
elrab |
⊢ ( 𝐴 ∈ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ↔ ( 𝐴 ∈ Cℋ ∧ ∪ { 𝐴 } ⊆ 𝐴 ) ) |
10 |
7 9
|
sylibr |
⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ) |
11 |
|
intss1 |
⊢ ( 𝐴 ∈ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } → ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ⊆ 𝐴 ) |
12 |
10 11
|
syl |
⊢ ( 𝐴 ∈ Cℋ → ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ⊆ 𝐴 ) |
13 |
|
ssintub |
⊢ ∪ { 𝐴 } ⊆ ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } |
14 |
4 13
|
eqsstrrdi |
⊢ ( 𝐴 ∈ Cℋ → 𝐴 ⊆ ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } ) |
15 |
12 14
|
eqssd |
⊢ ( 𝐴 ∈ Cℋ → ∩ { 𝑥 ∈ Cℋ ∣ ∪ { 𝐴 } ⊆ 𝑥 } = 𝐴 ) |
16 |
3 15
|
eqtrd |
⊢ ( 𝐴 ∈ Cℋ → ( ∨ℋ ‘ { 𝐴 } ) = 𝐴 ) |